For a vector bundle $E$, we have the structure equation relating a curvature form $\omega$ and its connection form $\Omega$, which is
$$\Omega=d\omega +\omega \wedge \omega$$
Under local trivializations, $\omega$ is a matrix with each entry a 1-form. Then how is $\omega \wedge \omega$ computed? Is it just the usual matrix product with multiplication on each entry becoming the wedge product?
Also, what does the notation $\omega^2$ mean? (I've seen it in some books. This also gives a matrix of two forms but how is it computed exactly?)