Currently there are only three known quests that directly involves skinning:
You need to have the Cenarion Beacon to get Patch of Tainted Skin via skinning from Arathandris Silversky (Alliance) or Maybess Riverbreeze (Horde) in southern Felwood after completing  Cleansing Felwood.
The quest is simple. You turn in 5x Patches of Tainted Skin for 2x Cenarion Plant Salves to the appropriate faction NPC. You get 4200 XP the first time you do it.
This is one of the daily quests on Netherwing Ledge where you turn in 35 Nethermine Flayer Hide and gain reputation with Netherwing and some cash.