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Gintama is the closest you'll get to Sket Dance, since Sket Dance's mangaka was an assistant on Gintama and emulates Sorachi's style.
Gintama is set in a feudal Japan that was conquered and modernized by aliens, and follows the story of Sakata Gintoki, a former samurai, who, together with his two friends and co-workers, takes on every job he can get in order to pay his rent.
Like Sket Dance, Gintama mainly focusses on comedy, but gets serious once in a while, with tragic pasts that haunt the characters or simply new developments that stray from the usual comedy approach of the show. However, the drama in Gintama's serious arcs is very action-oriented, so it may not be 100% what you're looking for. Still, you should give it a try.
The first few episodes weren't that great imho, but the show improved quite a lot, and now Gintama is one of my favourite anime.
You may know some characters from Gintama since there was a cross-over special with Sket Dance where characters from both shows met.
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