Hm, if that applies to tools as well, then the answer is actually no, as that would mean a Fine broad knife is no better than a Decent one for skinning. You get a fighting impact bonus, but no fighting skill bonus.
It can also be noted that the quality/type of tools can prevent you from producing items of high quality. If I remember correctly (which might not be the case), a broad knife worse than Decent is incapable of producing Superior skin, while a Decent knife of any other kind isn't sufficient to reach the Superior level (I think Fine is OK, but it might require Masterwork).
Acolyte posted while I typed...
See above. You should definitely get your hands on a Broad Knife because it's the right tool for skinning. Upgrading a Decent one to a Fine one may or may not achieve anything, however (you can also chase Njerps, as they frequently carry broad knives).