I have $N$ 1-forms $\omega_1(x), \ldots, \omega_N(x)$. I want to know if there exists an invertible linear combination of these forms which yields $N$ closed forms. In other words: does an invertible matrix $M(x)=(m_{i,j}(x))_{i,j}$ exist, such that the forms $\eta_i = \sum_{j=1}^N m_{i, j}(x) \cdot \omega_j(x)$ fulfill $d \eta_i(x) = 0 \, \forall \, i=1, \ldots, N?$
I am pretty sure that the necessary and sufficient condition for such an $M$ to exist is the "Frobenius condition"
$$ d \omega_i \wedge \omega_1 \wedge \ldots \wedge \omega_N = 0 ~ \forall ~ i=1, \ldots, N. $$
However, I am looking for a good source of this statement.