Bike Trainer - HELP!

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I bought a McLain bike trainer from a friend, took it home, and it mounted easily to the frame. I easily slid the silver cylindrical clamp over the nut on the chain side.

However, on the bike's left side there's a skewer and the clamp won't fit over it. The best I can do is tighten the trainer against the skewer, but it rubs and scratches. It seems steady, but I know that can't be how it's supposed to work!

Has anyone had this problem? Do I have to remove the skewer?

THANKS!!