Let $\nabla$ be a connection on a vector bundle $E$. Suppose $\omega$ and $\Omega$ are the connection and curvature matrices of $\nabla$ relative to frame for $E$ over an open set. Then for any integer $k \ge 1$ $$ d(\Omega^k) = \Omega^k \wedge \omega - \omega \wedge \Omega^k. $$
The case for $k =1$ is easy since we can differentiate the second structural equation to get
\begin{align*} d\Omega &= d(d\omega) + (d\omega) \wedge \omega - \omega \wedge d\omega \\ &= d\omega \wedge \omega - \omega \wedge d\omega \\ &= (\Omega - \omega \wedge \omega) \wedge \omega - \omega \wedge(\Omega - \omega \wedge \omega) \\ &= \Omega \wedge \omega - \omega \wedge \Omega. \end{align*}
But I find it difficult to prove for larger $k$. I thought induction would do it, but for the life of me I cannot find anywhere how to manipulate matrices of differential forms. What I currently have is the following.
Suppose that the equation holds for $k = n$. We have that \begin{align*} d\Omega^{n+1} &= d(\Omega^n \wedge \Omega) \\ &= d\Omega^n \wedge \Omega + (-1)^n\Omega^n \wedge d\Omega \\ &= (\Omega^n \wedge \omega - \omega \wedge \Omega^n) \wedge \Omega + (-1)^n\Omega^n \wedge d\Omega \\ &= (\Omega^n \wedge \omega - \omega \wedge \Omega^n) \wedge \Omega + (-1)^n\Omega^n \wedge (\Omega \wedge \omega - \omega \wedge \Omega) \end{align*}
Simplifying the last expression is not that clear to me.
There is no $(-1)^n$. Remember that $\Omega$ is a matrix of $2$-forms. Simplifying what you have if we removed that $(-1)^n$ does it: $$\Omega^n\wedge\omega\wedge\Omega - \omega\wedge\Omega^{n+1} + \Omega^{n+1}\wedge\omega - \Omega^n\wedge\omega\wedge\Omega = \Omega^{n+1}\wedge\omega - \omega\wedge\Omega^{n+1},$$ as required.