Let $\omega$ be an 1-form and $X$ be a vectorfield. As usual $i_X \omega$ denotes the interior product and $\mathrm d$. the exterior derivative.
Is there an expansion of the term $$ \mathrm d (i_X \omega) \quad ? $$
I would suspect that this can be expressed in terms of Lie-Derivatives, wedge-products and $i$ and $\mathrm d$ but I could not find anything in the literature.
Cartan's formula tells you that $\mathcal{L}_X \omega = di_X \omega + i_X d\omega$ (for forms of arbitrary degree).
If $\omega$ is a $1$-form, then $\iota_Xd\omega(Y) = d\omega(X,Y) = X\omega(Y) - Y\omega(X) - \omega([X,Y])$, hence you can further write $$ di_X \omega(Y)=[\mathcal{L}_X,i_Y] \omega +\omega([X,Y]) + Y\omega(X) $$