It appears that this problem happens randomly. Sometimes I can skin without problem and sometimes it doesn't work until server restarts again.
Edit: It seems animals can't be skinned using any bladed melee whatsoever. Machete, kitchen knife etc. doesn't work as well.
Edit2: The problem is DEFINITELY related to persistance. In any non-persistant server, everything works like a charm. However, in persistant servers, mechanics such as skinning, crafting splint, handcuffing, force-feeding, pouring water from bottle to bottle are broken. I hope this will be fixed by the next patch.