A class of cohomology

Question

Let $(E,g)$ be metric vector bundle over a manifold $M$ with a connection $\nabla=d+A$. Then if we define:

$$\alpha=tr(A)$$

It is a 1-form because if $h\in SO(n)$, $tr(h^{-1}dh)=tr(dh^* h)=-tr(h^{-1}dh)=0$. Moreover, we have:

$$d\alpha=0$$

because $R=dA+A\wedge A$ and we suppose $tr(R)=0$.

We obtain so a class of cohomology. Is it an invariant of the vector bundle $E$?