Let $\mathfrak{g}$ be a Lie algebra and $\rho:\mathfrak{g}\longrightarrow \mathsf{End}(V)$ a representation of $\mathfrak{g}$ on a vector space $V$. How is the Chevalley-Eilenbeg differential $$d_\rho:\Lambda^p \mathfrak{g}^* \otimes V\longrightarrow \Lambda^{p+1}\mathfrak{g}^*\otimes V$$ defined? I'm looking for the definition without using the identification $\Lambda^p \mathfrak{g}^*\otimes V\simeq \mathsf{Hom}(\Lambda^p \mathfrak{g}, V)$. Any reference would also be welcome.
I know that using the previously mentioned identification $d_\rho$ is given by:
$$(d_\rho \varepsilon)(x_1\wedge \ldots \wedge x_{p+1})=\sum_{j=1}^{p+1} (-1)^j \rho_{x_j}(\varepsilon(x_1\wedge \ldots \wedge \widehat{x_j}\wedge \ldots x_{p+1}))+\sum_{i<j} (-1)^{i+j} \varepsilon([x_i\wedge x_j]\wedge x_1\wedge \ldots \widehat{x_i}\wedge \ldots \wedge \widehat{x_j}\wedge \ldots x_{p+1})),$$
but for my purposes I really need to know on elements of the form $\varepsilon\otimes v$.
Indeed, is there an axiomatic caracterization of $d_\rho$ like it happens to the de Rham differential? It should be the unique derivation of a certain product on $\Lambda^p \mathfrak{g}^*\otimes V$ satisfying some properties?
Thanks.