If we denote by $\nabla$ the connection on $E\to M$, then we can define an exterior differential $d^\nabla:\Gamma(\Lambda^pM\otimes E)\to\Gamma(\Lambda^{p+1} M\otimes E) $ by $$ d^\nabla \alpha (X_0,\dots, X_p) = \sum_i (-1)^i \nabla_{X_i}(\alpha(\tilde{X_0}, \dots , \hat {\tilde{X_i}}, \dots, \tilde X_p)) + \sum_{i\neq j} -(1)^{i+j} \alpha ([\tilde X_i, \tilde X_j], X_0, \dots, \hat X_i,\dots, \hat X_j, \dots, X_p).$$
where $X_i\in T_x M$; $\tilde X_i$ denotes an extension of $X_i$ to a neighbourhood of $x\in M$, and the hat above something denotes that that argument has been omitted.
This formula can be found in Besse's book "Einstein Manifolds" pg 24, beware I recall there are a few typos in that part of the book.
The pattern for this definition is the usual one used to extend the covariant derivative to the tensor algebra, modified by alternating the result and using the covariant derivative.