I must disagree with Andrew on his claim that in AmE "media" is definitely singular.
Looking at the CoCA corpus "media have" and "media has" have roughly equal counts. I'm not a professional researcher and my ability to use the byu corpora is limited but just going through the results you get examples such as
National media have noticed (Denver News)
the news media have reported (NYtimes)
And the public and the media have the right (spoken CNN)
There is a higher incidence of "media has" it seemed going by a rough look but both versions are certainly present even in AmE.
The oxford dictionaries say:
The word media comes from the Latin plural of medium. The traditional view is that it should therefore be treated as a plural noun in all its senses in English and be used with a plural rather than a singular verb: the media have not followed the reports (rather than ‘has’). In practice, in the sense ‘television, radio, and the press collectively’, it behaves as a collective noun (like staff or clergy, for example), which means that it is now acceptable in standard English for it to take either a singular or a plural verb. The word is also increasingly used in the plural form medias, as if it had a conventional singular form media, especially when referring to different forms of new media, and in the sense ‘the material or form used by an artist’: there were great efforts made by the medias of the involved countriesabout 600 works in all genres and medias were submitted for review.
All in all both plural and singular media is correct in current usage.
They are both grammatically valid, but you have to be aware of the logical context. They are both correct from a strict grammatical perspective, but the first one makes more sense and sounds much more natural in this particular construction. In the context of a comparison ("is better than") the reader needs to know what is being compared. When you use the plural, it sounds like you're comparing one group to another, and it is not clear what the group is.
Typically there is only one man and one woman who would be in a relationship at any given point in time, so the singular sounds better. When you say "no men are better than women at maintaining relationships", it implies there are multiple men involved in the type of relationship in question. Unless the context specifically relates to same-sex relationships or group relationships only, it's confusing and doesn't seem to fit the logic of the proposition.
There might be instances where comparing groups like that might make sense. For example, you could say "no Americans are better than Canadians at fielding a hockey team". The literal implication is that you could select any group of Americans and any group of Canadians and that would always be true, so probably it's logically false, but because people play hockey in groups it makes a little bit more sense. It's the sort of thing a drunk Canadian might say instead of "Canadians are usually better at hockey than Americans."
Where "no men" makes more sense. Outside of the comparison context, it is easier to find circumstances where "no men" makes sense and would be the more natural usage. For example, "no men are allowed into the women's restroom" sounds good and it is clear that you're referring to all men everywhere.
In general the construction "no man" sounds a bit like a proclamation. It is not unheard of for ordinary use, but perhaps more something a king or philosopher would say. So the phrasing "no man is allowed into the women's restroom" is fine in a technical sense, but sounds a little bit like a sign on a royal bathroom (or an English language learner). The phrasings "no man has set foot on Mars" and "no men have set foot on Mars" are both equally valid and natural. Because going to Mars is a grander undertaking, the more dramatic phrasing of "no man has set foot on Mars" becomes more appropriate than when you are discussing a bathroom.
Best Answer
If each individual [item] implied by a plural subject has only one of something (each of us has one face, in OP's example), we tend to extend the plurality of the subject (we) to the object (faces). Thus:
But if you look at some of the results there, I'm sure you'll find that the second (singular) set will have more contexts where we're all taking pride exactly in the same thing (as opposed to each of us being proud of our specific version of whatever it is).
So in OP's specific example, most people in most contexts would use plural faces. But if preceding text had focused on "the face" in a more "generic, archetypal" way (for example, A stranger's face is usually the first thing we look at when we meet them), it wouldn't be unreasonable for the writer to continue with the singular when recommending that everyone should wash their face properly.
There's one additional point to consider in contexts like OP's. Sometimes (especially when addressing children), we use we when we really mean you (singular or plural, we do it in both contexts). Thus:
That interpretation (we means you, and doesn't include the speaker) isn't always possible. But it certainly could apply in OP's context, so bearing in mind this would represent a very condescending style of address, that might be one more reason to use the plural in that exact context.