If it was physically capable of speaking, it would be able to speak the languages it knew in life. It still retains its knowledge of them, so knowing how to speak a language is not the problem. The problem is that a skeleton lacks lips, a tongue, vocal cords, a voicebox, and lungs. Speaking is simply impossible.
The Monster Manual entry on skeletons says:
They can't read, speak, emote, or communicate in any way except to nod, shake their heads, or point.
From this, it seems clear that it is meant to be impossible for skeletons to communicate in any way other than the given nodding, shaking, and pointing.
Drow sign language isn't an officially included language in D&D 5e rules, at least not as far as I can find.
The languages a PC knows are determined by their race and, save for a special few, are mostly listed on page 123 of the PHB. You'll notice that's there's no mention of a drow sign language. It's not even mentioned in the Sword Coast Adventurer's Guide (a 5e sourcebook for campaigns set in the Forgotten Realms).
If you also look at the references for the Wikia page you've linked you'll notice that it's mostly referencing novels by R.A Salvatore set in the Forgotten Realms and previous editions of D&D. In general, a novel does not equal a rules books for a TTRPG. Even the other question from this site that you've linked is asking about 3.5e, not 5e.
In terms of how much influence a novel does have on rules created for a TTRPG, well, that will be determined by your DM and the world they set their campaign in.
However, this doesn't mean your drow can't know drow sign language. It just means that, by default, it doesn't know it because it's not a part of the rules.
A DM is free to create their campaign setting, their world, their universe, their multiverse, however they please. If you want your drow the know sign language, talk with the DM; see if it's something they think would make sense in their world and if they can incorporate it.
The DMG even has a section on languages, from page 20-21:
When fleshing out your world, you can create new languages and dialects to reflect its unique geography and history. You can replace the default languages presented in the Player's Handbook with new ones, or split languages up into several different dialects. [...] You might invent additional secret languages, besides Druidic and thieves' cant, that allow members of certain organizations or political affiliations to communicate.
Best Answer
Yes, RAW others can learn.
As always DMs can rule however they want and may require you to find a willing drow to teach you. (If I allowed it, I would require this.)