Yep.  Never waste anything, if possible.  Return it to earth.

Same applies to trapping muskrats.  Keep the pelts, 'donate' the meat.

I think it's okay to keep trophy's (a rug or mounted in-motion) if a bobcat or bear was in the act of attacking me or my party. 

Never from hunting.   Nor from fishing either, especially in these days of phone-cams.

Though I might be tempted to keep an adult sturgeon as a trophy-piece.