The title says it all: We now consider differential forms which are also vector fields, which are called vector-valued differential forms. The standard example is $d\rr$ itself, which is both a 1-form and a vector field. More generally, a vector-valued $p$-form can be written as $\alpha^i \ee_i$, where each $\alpha^i$ is a $p$-form, and where $\{\ee_i\}$ is a vector basis (here chosen to be orthonormal).