A closed $1$-form in a simply connected set in $R^n$ is exact. I would like a similar condition (with a reference) on sets in $R^n$ that closed $2$-forms are exact. De Rham cohomology gives an algebraic answer, but I am interested in a condition like simply connected with geometric appeal.
The condition should not exclude "too many" sets. By analogy, closed $1$-forms on contractible sets are exact. But this condition excludes too many sets, e.g., the simply connected set $R^3 - \{{0\}}$.