Since patch 1.9 there is no maximum level anymore.
If you have a skill at 100 you can make it "Legendary". This resets the skill to 15, gives all perk points invested in that specific tree back. Be warned: You lose all the benefits of having a higher skill level! The advantages of making skills legendary are:
- Difficulty increase, since the enemies don't have their skills reduced.
- You can continue to level up and earn extra skill perks.
It is possible to repeatedly make a skill legendary or to make multiple skills legendary, this allows you to (slowly) unlock all perk points.
Level 81 is the maximum level in Skyrim without making any skill Legendary
There is a linearly constant amount of "experience" required for each level. The formula is simply:
(level-1) * 25 + 100
Where level is your current level.
"Experience" in this case is earned strictly through skill ups. Leveling a skill to X will give you X experience towards your next level. For instance, to level from level 1 to level 2, you require 100 experience. This means you can either raise 5 skills to level 20, one skill to level 21 (assuming it starts at 15), or any other permutation thereof.
Also, and this is important:
There is no soft cap
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Level 81 is just as (relatively) easy to reach as Level 11, it just requires more total experience.
Since all races have +10 to one skill and +5 to five skills, this means there is a total of 88,085 "experience" to earn, and this is constant across the ten races. Level 82 would require 89,100 total experience, and is currently unreachable.
Thus, when your character maxes all of their skills, they will be level 81.
As you noticed, Destruction is pretty straightforward. Each cast and while continued casting, you will increase skill, but this has a caveat: only if you are attacking someone. If you're fizzling your spell in the middle of town, it's not going to increase it.
Alchemy is also straightforward: each potion created will increase your skill. As dpatchery notes, eating ingredients for effect determination will also increase your skill by a small amount.
With Enchanting, each enchant (or disenchant, as LessPop_MoreFizz notes) as will increase skill. dpatchery also notes that you can increase your skill by recharging items with soul gems.
Restoration, namely the healing spells, only increase skill if your health is below its maximum. The others only work if you're in combat or near hostile enemies: this includes spells like Steadfast Ward.
For summoning spells (essentially most things in the Conjuration school), you won't get a skill up for the summon until you enter the range of hostile mobs and, in the case of summoned creatures (like the Atronach and the Familiar), they do some damage. With bound weapons, entering range of hostile mobs is all that's necessary to gain Conjuration skill; however, using them in combat will increase their appropriate weapon skill, not Conjuration.
Alteration and Illusion spells that are targeted towards messing with hostile mobs need hostile mobs within range to skill up. Buffs, like Oakflesh, don't increase skill without hostility as well. Others, like Candlelight and Muffle, are recastable without hostility but only increase skill by a small amount.
Best Answer
You must first max out the skill (reach 100) then you will have the option to make the skill legendary on the skill tree screen. You can see more on this YouTube video.
Yes you can.