The way I look at it is that these animals are here in the lab for me to learn from; so that I can better serve that species in the future. I want to learn all I can from these dissetions so that in future, doing surgery, I will feel more comfortable and maybe save a life (or at least not take one.) We have already done a dog, a cat, and are currently working on the horse. You can frequently find me petting the animals and I even tried to "restrain" our horse at one point. I can't escape the fact that these are real animals, no matter how much anyone tells me to put that part of it out of my mind. The dissections really don't bother me though, because like I said, it's for a future, greater good.
So personally, I say you don't need to lose that emotional connection to the species. I'm a dog person, and a huge horse person and I don't think I could really separate my emotions even if I wanted to. One of the horses we are dissecting looks like a mini version of my old gelding but even that I'm okay with, as long as everything is treated respectfully.
I have noticed that as we get farther and farther into an animal (and it looks less and less like its alive counterpart) that it is easier to forget though. Not to be gruesome, but currently, our horses no longer look like horses...
When skinning, just take a deep breath. Remember, if you know the feel of the skin and all that, your future patients will greatly appreciate that knowledge if it means a less traumatic surgery for them.
Edit: And something else that really helped was being able to go home at the end of the day and relate all the things we learned in anatomy lab to real life. IE, palpating different muscles, seeing which bones are palpable, etc, on a real animal. It really helps the whole point of doing this sink in.