Let $M$ be a smooth manifold, $E\to M$ a smooth vector bundle equipped with a linear connection $\nabla$. Then we have the exterior covariant derivative $$\nabla^2:\Gamma(M,E)\to\Gamma(M,\Lambda^2T^*M\otimes E).$$
If moreover $M$ has a Riemannian metric and thus $T^*M$ has an induced Levi-Civita connection, then we can form the second covariant derivative $$\nabla\nabla:\Gamma(M,E)\to\Gamma(M,T^*M\otimes E)\to\Gamma(M,T^*M\otimes T^*M\otimes E).$$
My question is:
How are these two operators related?
For example, $\nabla^2$ is in fact the alternating part of $\nabla\nabla$. Although this can be proved by direct calculations (i.e. they are both the curvature $2$-form), but why? Is there any intuition for this? Furthermore, it is strange that $\nabla\nabla$ actually depends on the Riemannian metric, whereas its alternating part does not.