Hammarbybacken is in stockholm, never skied there but its not very big, but from what I know, its skiable. There are a few other places to ski in stockholm. Here's a map you can use to locate skiresorts http://www.freeride.co.uk/ski-resorts/
Åre can get very crowdy, especially during the holidays. There are a lot of ski resorts here in sweden, and if you have the opportunity to bring skis I would recommend to bring them with ya.