The Chern-Simons 3-form is given by

$\omega_3={\rm Tr} \left[ A\wedge dA+\frac{2}{3}A\wedge A\wedge A\right]$

where $A$ is a connection one-form in the adjoint representation of a non-Abelian gauge group.

My differential geometry is rather rusty (and this is new to me too) hence my questions;

$A$ is a 1-form. By definition of the wedge product between a $p$ form $\alpha$ and $q$ form $\beta$ we have $\alpha\wedge\beta=(-1)^{pq}\beta\wedge\alpha$. So we should have $A\wedge A=-A\wedge A=0$.

Why is this not the case?

Next question; I want to calculate $d\omega_3$ Does the fact that everything is inside the trace effect my calculation? In other words does the differential operator pass through the trace and only act on the forms?