To go straight to the point I am dealing with a problem that requires me to integrate the orbits of 5-form, that is, finding its null quadri-directions which satisfy, $\omega(X)=0$ where $\omega$ is a 5-form constructed by wedging five 1-forms.
The $X$ is a quadri-tangent which has the form, $X=X_{\mu}\otimes X_{\nu}\otimes X_{\rho}\otimes X_{\sigma}$ and thus I find myself trying to make sense of some like this, $\left(\alpha\wedge\beta\wedge\gamma\wedge\lambda\wedge\sigma\right)(A\otimes B\otimes C\otimes D)=?$, where $\alpha$, $\beta$, $\gamma$, $\lambda$, $\sigma$ are 1–forms and $A$,$B$,$C$,$D$ are vectors.
Anyone has an idea of the exposes decomposes in term of contraction between underlying 1-forms and vectors?