If you're skinning it to sell to the fur trade, just keep the fur on the back of the ear and cut the cartilage and inside fur off. If you're going to tan it and hang it on the wall, separate the rear skin from the cartilage (be careful at the edges), trim off any meat and sinew on the cartilage, and leave it all (rear fur, cartilage, and inside fur) attached. It is possible to remove the cartilage and leave the front and back skin, but it's not easy and even then there'll likely be holes.
Just noticed your note on the stretcher. If you're going to tan it, don't put it on the stretcher. Once you've got it skinned you should flesh it and then salt it. I don't know what method you're using to tan it, but just drying it on a stretcher won't let you keep it for very long. Eventually it'll start to smell and some of the hair might fall out. Most hides that are being tanned are salted first and then put into a pickling brine. From there it's removed and allowed to dry. Then you'll have to break it down (the hide will be somewhat stiff). I've heard of people running the pelt back and forth in a rope noose to help break down the stiff pelt. The last step is usually to rub in some neatsfoot oil and then work the hide again until it's nice and soft. It sounds like a lot of work to me, but I'm sure you'd have something to be proud of knowing you did it yourself. Tanneries don't charge too much to do a coyote....$50 or so. Good luck with it! Let us see the finished product.
