This is probably a stupid question, but let say we have a complex manifold $S$ and a matrix of holomorphic 1-form $\omega_{i,j}$ and we would like to find a holomorphic $a : S \to \mathbb{C}^n$ to the differential equation $ da=\omega a$. I found on a book that necessary and sufficient condition for existence of a local solution a is the integrability condition $d\omega + \omega \wedge \omega=0$.
I can see why it is necessary (just using $d(da)=0$),but where the sufficiency comes from ?