I don't think it should be a skill but i would love to see this implemented into the game. Like you said it will give us a variety and fish will not be the top 10 healing items.
So Cows would drop a carcass which could be cut up with say a Butchers Cleaver and then could be cooked by a person who wishes to train cooking awesome!
This seems like a good idea although it is too close to cooking, and i feel like it could be like making a fire with a bow in Runescape, you get firemaking and crafting XP.
So Dragons drop instead of skin + bones just a carcass- you use a knive on that carcass, then you would get skin/bars + bones + dragon meat + butcher xp? Like that idea
Although not all monsters should drop carcasses(f.e. no humanlike beings).