Let $(\mathcal E,\pi,M)$ be a G-vector bundle with a linear connection.
On the associated principal bundle, it is true that $\Omega=d_\omega\omega$, where $\Omega$ is the curvature form, $d_\omega=d\circ h^*$ is the covariant exterior derivative and $\omega$ is the $\mathfrak g$-valued connection form.
On the other hand, if we consider the vector bundle $\mathcal E$ only, if $\psi$ is a section, we can describe its covariant derivative locally as $$ d_\omega\psi=d\psi+\omega\psi. $$
Here $\omega$ is a local Lie-algebra valued 1-form. If we ignore its pathological transformation properties under the change of a local trivialization, we may see $\omega$ as either a local section of the adjoint bundle (adjoint bundle as in $P\times_{Ad}\mathfrak g$ where $P$ is the associated principal bundle), or as a local section of $\mathcal E \otimes\mathcal E^\ast$.
The action $\omega\psi$ is understood naturally if we go with the $\mathcal E \otimes\mathcal E^*$ view, otherwise if $G$ acts on the model fibre $E$ via a representation $\rho$, we may understand the action $\omega\psi$ to be happening through the corresponding Lie algebra representation $d\rho_e$.
We may extend the covariant exterior derivative to an $\mathcal E$ valued $k$-form as follows: If locally $$ \psi=\sum_{\mu_1<...<\mu_k}\psi_{\mu_1...\mu_k}dx^{\mu_1}\wedge...\wedge dx^{\mu_k}, $$ then $$ d_\omega\psi=\sum_{\mu_1<...<\mu_k}d_\omega\psi_{\mu_1...\mu_k}\wedge dx^{\mu_1}\wedge...\wedge dx^{\mu_k}. $$
We may also extend $d_\omega$ to $k$-forms which take their values in the tensor product bundle constructed out of $\mathcal E$ and $\mathcal E^*$.
Particularily, for a $\mathcal E \otimes \mathcal E^*$-valued $1$-form $\lambda$ we have $$ d_\omega \lambda=d_\omega\lambda_{\mu}\wedge dx^\mu=(d\lambda_\mu+\omega\lambda_\mu-\lambda_\mu\omega)\wedge dx^\mu=d\lambda+(\omega_\nu\lambda_\mu-\lambda_\mu\omega_\nu)dx^\nu\wedge dx^\mu=d\lambda+\omega\wedge\lambda+\lambda\wedge\omega.$$
If we apply this to $\omega$ itself, we get $$ d_\omega\omega=d\omega+2\omega\wedge\omega, $$ on the other hand, the curvature form is $$ \Omega=d\omega+\omega\wedge\omega. $$
So questions:
Is the curvature actually the covariant exterior derivative of the connection? Doing this on the principal bundle says yes, the derivation here says no.
If so, how come I get that factor of 2 here? Did I make a mistake or what? Is the covariant exterior derivative of an adjoint-bundle valued form not the same as the covariant exterior derivative of an $\mathcal E \otimes \mathcal E^*$-valued form?