Let $F \in L^{p}(\mathbb{(-1,1)^{n}}; \Lambda^{2}\mathbb{R}^{n})$ be an $L^{p}$ $2$-form on the open cube $(-1,1)^{n}.$ Suppose we know that for every $x \in (-1,1)^{n},$ that there exists $0 < r = r(x) < \operatorname{dist}(x, \partial (-1,1)^{n})$ such that $$ dF = 0 \quad \text{ in the sense of distributions on } B_{r}(x). $$
Does it imply $$ dF = 0 \quad \text{ in the sense of distributions on } (-1,1)^{n}? $$
I think it must be true but somehow I can not figure out a way to prove it.