recently bought a second hand cycleops fluid 2 trainer that doesn't come with a skewer. However, the one that is one my bike looks almost identical with the cycleops one, wonder what your's looks like when it is on the trainer. Mine looks like this when it is on close position SDC11038.jpgSDC11037.jpg thank you
It goes in deeper, but I don't think it makes much difference in practice. Neither the skewer nor the trainer arm are going to spin. The important part is that the diameter of the skewer closely matches the diameter of the opening in the trainer, so that the bike does not wobble when you're pedaling.
You can go to your LBS and get a trainer specific skewer for like 8 bucks. They are thicker than the stock skewers that come with your bike/wheels. That's what I did as I also got a used trainer and it didn't come with a skewer.