What is causing the chain skip after chain replacement?
Yes, the stretched chain wears the chain ring and sprockets to match its pitch. It's recommended to use a chain gauge (such as this one by Park Tools) so you can replace your chain before it's stretched enough to do significant damage to the rest of the drive train. If you do let the chain stretch too far, you'll need to change the cassette/rear cog and/or chain ring(s).

The teeth of new chain rings have a mostly symmetrical profile; worn ones acquire a characteristic "shark fin" shape that's steeper on one side than the other and quite pointy at the top. On a one-by or single-speed bike, that's pretty easy to identify; on a bike with front derailleurs, be aware that the chain rings tend to have several teeth with funny profiles from new, to aid with shifting.