Let $\omega_1$ and $\omega_2$ be differential $p$-form and $q$-form, respectively. Given $p+q$ vector fields $X_1, X_2, \dots X_{p+q}$, I am wondering that if there is a relation between $\omega_1 \wedge \omega_2 (X_1, \dots, X_{p+q})$ and some product of $\omega_1(X_{i_1}, \dots, X_{i_p})$ and $\omega_2(X_{i_{p+1}}, \dots, X_{i_{p+q}})$? For example, if $p = q =1$, we have simply \begin{eqnarray} \omega_1 \wedge \omega_2 (X_1, X_2)= \begin{vmatrix} \omega_1(X_1) && \omega_1(X_2) \\ \omega_2(X_1) && \omega_2(X_2) \end{vmatrix}. \end{eqnarray}
I have also tried the case $p = n, q = 1$ for $n \in \mathbb{N}$, in which I get \begin{eqnarray} \omega_1 \wedge \omega_2 (X_1, \dots, X_{n+1}) = \sum_{k = 1}^{n+1} (-1)^{k+1} \omega_1(X_1, \dots, \hat{X}_k, \dots, X_{n+1}) \omega_2(X_k). \end{eqnarray}
Have someone had any idea about the general case?
If $\omega_1$ and $\omega_2$ are $p$- and $q$-forms respectively, and $\{X_i\}$ is a set of $p+q$ vector fields, then the general result is, $$ (\omega_1 \wedge \omega_2) (X_1, \ldots, X_{p+q}) = \frac{1}{p!q!} \sum_{P\in S_{p+q}} \text{sgn}(P)\, \omega_1(X_{P(1)},\ldots,X_{P(p)})\,\omega_2(X_{P(p+1)},\ldots,X_{P(p+q)})\;, $$ where $S_{p+q}$ is the set of permutations $P$ of $p+q$ integers and sgn$(P)$ is the sign of the permutation.
See for example equation (5.65) in Nakahara's Geometry, Topology, and Physics.