So what properties should exterior differentiation of vector fields satisfy? Like all derivative operators, we must clearly demand linearity: \begin{equation} d(a\,\vv+\ww) = a\,d\vv+d\ww \end{equation} where $a$ is a constant, and a product rule: \begin{equation} d(\alpha\,\vv) = d\alpha\,\vv + (-1)^p \alpha\wedge d\vv \end{equation} where $\alpha$ is a $p$-form. An alternative version of the product rule is obtained by reversing the order of the factors: \begin{equation} d(\vv\,\alpha) = d\vv\wedge\alpha + \vv\,d\alpha \end{equation} which has no minus signs, since $\vv$ is a (vector-valued) 0-form.

Do these properties suffice to determine the action of $d$ on vector-valued differential forms? No! We need more information about $d$ acting on a vector basis.