Let $\omega \in \Omega ^k (M)$ and $\mu \in \Omega ^l (M)$ and the exterior derivative defined by
$$ d \omega (X_1 , \dots , X_{k+1}) := \sum _{i = 1} ^{k + 1} (-1)^{i+1} X_i (\omega (X_1 , \dots , \hat{X}_i , \dots , X_{k+1})) \\ + \sum_{i<j}^{k+1} (-1)^{i+j} \omega ([X_i , X_j] , X_1 , \dots , \hat{X}_i , \dots , \hat{X}_j , \dots , X_{k+1}). $$
How can I prove that
$$ d (\omega \wedge \mu) = d \omega \wedge \mu + (-1)^k \omega \wedge d \mu $$
without using a local representation (in some chart) of the differentials forms?