If $\alpha$ and $\beta$ are closed forms on the open unit ball, is there a simple way to show that $\alpha \wedge \beta$ is exact without resorting to the full "closed form is exact on a star convex set" proof?
Well, $\alpha$ and $\beta$ being closed only tells you $\alpha \wedge \beta$ is closed. To deduce anything about the exactness of that, you at least need e.g. $\beta$ is exact. So you'd want to prove that a closed form on a ball is exact.
If you try to write down a proof I don't think that can differ much from the star convex set proof you have in mind.
I hate to admit it (since Balarka has only known this for a day), but Balarka is right. There's nothing special about the wedge product. You need the general result.
BTW, Balarka, that result about star convex sets will be an exercise in the last section. You can try to understand that there's a chain homotopy there.
