Have done this, but the golf ball method as per
@dodgyknees seems like the way forward from now on!
Guessing you don't neck shoot, or rate the silversides, half of which were left with the skin?
Hang to mature, with the head on until ready to skin (stretches the neck so it is straight rather than bent round); using a rounded nosed tripe-knife or roeing knife, slit under the skin of both front legs from inside of leg and forwards toward the gullet, as opposed to straight up toward the brisket (you'll get on cleaner and easier this way);
Point of knife indicating exit point of first incisions made along the inside of each foreleg;
make an incision across the inside of the meaty bit of the foreleg, a bit like cutting its wrists, about 3/4" above the joint where the foreleg was removed (to give your loops something to bite into, without wasting meat as such, it's all tendon nearest the joint);
Slitting the 'wrist' so the loop can 'bite' in, little wasted meat!
hang up about 8-9' off the ground, and begin to pull down the way on the pre loosened skin of the lower 'arm' of the foreleg; once you come to the shoulder area, you may want to stop there, and pinch hold of the tapered strip of skin left between the pit of the shoulder and the gullet, this will peel back pretty easily.
Some care is required both as you leave the shoulder and peel down either side of the split brisket, and also when going forward to free the neck, as the meat there is fairly tender and can tear, this can be avoided by running your knife more or less parallel about a centimetre in on to what looks like the neck meat, but is actually the soft tissue surrounding the jugular vein running along either side of the neck (not recommended eating, so no waste in doing this); the head can be freed at the atlas joint along with the skin, with minimal hair left on the carcass neck.
The rest should pull down fairly easily, but like everything in life, practice makes perfect, there's no great rush, and better to have something clean and hair free at the end, which after all is the objective of the exercise. If the deer has a tail this will have to be severed at the root, which is readily enough achieved through the joint. The haunches can be barrel skinned, ie no knifework really necessary, but you ay wish to go very carefully when pulling the skin off the upper thigh/hip area, so as not to tear the silversides, this is where judicious use if the knife to seperate the connective tissue from the meat is worth considering.
Done in this way, you are pulling always down and away from the clean carcass, and with a modicum of care you can end up with a hair- and fingerprint-free result.
Showing the cuts made from inside of foreleg to gullet; the back legs are more often than not hanging down toward the floor at this point, rather than still hanging up;
- a little tricky in the 'best practice triangle' (here unmolested!), but otherwise plain sailing to this stage; all loose skin and hair with debris falling down and away from the edible bit
Head removed in the thumbnail below, some more pics next post, 5 file limit...