Consider (locally) the following system of linear first order PDEs: $$ \forall\,1\leq j<k\leq n,\quad \frac{\partial F_k}{\partial x_j}-\frac{\partial F_j}{\partial x_k} = \nu_{jk}(x), $$ for the unknown functions $F_i(x_1,\ldots,x_n)$, and source terms $\nu_{jk}(x_1,\ldots,x_n)$.
Question: Give conditions for the existence of smooth solutions $(F_1,\ldots,F_n)$.
An idea is to apply Frobenius theorem, but for that one should split the system in separate equations for each unknown. I do not succeed in finding such splitting.
Solution: Based on comment of Ted and Ivo, the necessary and sufficient condition is simply $d\nu=0$ where $\nu=\sum_{k<j}\nu_{k,j}dx_k\wedge dx_j$ is a differential 2-form.