First, let me ask this. The gear where the skipping is worse, is that the gear you use most of the time?

If so, then yes, your cassette is worn out as well.

What happens, is that as a chain wears, the overall length increases (bigger and bigger gaps at the pins, rollers, and links), well, as the chain lengthens, the teeth on the cassette wear equally.

If the new chain shifts well in the gears that are not worn as much, but skips in the gears that were ridden most of the time, you can bet you waited too long to replace your chain and now have to replace the cassette as well.

I'd suggest also buying one of these:http://www.parktool.com/tools/CC_2.shtml