You answered the question yourself. Skin it just before you cut it up. I skinned a buck out once and didn't get around to cutting it up for a couple of days. It had dried out a lot in that time. I never did that again. I've never found that hanging a deer for any length of time improved the quality or flavor of the meat. It just dries out. Some of the best eating deer were the ones that were cut, wrapped and frozen the day they were shot. I find that getting as much fat off the meat as possible works for improving the flavor.