The advice above is spot on about chain and cassette wear and how new chains on worn cassettes cause skipping. If you still have the old chain, check it for stretch, which will give you a good indication of the condition of the cassette.

If you don't have the old chain you can check the cassette condition by one of these two ways.

1- shift into the problem gear, and use an ice pick or small screwdriver to gently lift the chain away from the sprocket at the mid point of the chain wrap.
with a new chain and cassette it'll barely lift, but with wear there's movement possible so it'll lift and the chain will shift over on the nearby teeth to allow it. If you can lift more than about 3/16" you're in bad shape.

2- requires a friend. Lean the bike in a corner with the front wheel against the other wall. Shift into problem gear, and have your friend stand on the pedal to load the chain while you carefully watch the cassette. New chains and sprockets will not move but with chain and/or sprocket wear, the chain will shift forward and climb higher on the sprocket teeth, if worn enough it'll climb out entirely causing the skip.

Note both tests will tell you that either the chain or sprocket is worn, but not which. In your case, since you know the chain is OK, you know it's the cassette.