Let us consider a current on $\mathbb R^n$, that is a differential form whose coefficients are distributions. For simplicity, let us check the case of a $1$-form $$ u=\sum_{1\le j\le n} u_j dx_j,\quad u_j\in \mathscr D'(\mathbb R^n), $$ and assume that $du=0$, i.e. $\partial u_j/\partial x_k=\partial u_k/\partial x_j$. I want to prove that there exists $a\in\mathscr D'(\mathbb R^n)$ such that $da=u$. The same question can be raised for tempered distributions and also for higher degrees.