Consider the form $2p+1$-form given by: $\omega_A = A \land \mathrm{d}A$ for a certain $p$-form $A$. It easy to prove that it is a closed form: $\mathrm{d}\omega_A = \text{d}A \land \mathrm{d}A + (-1)^p \mathrm{d}^2A = 0$.
Assuming that $A$ is defined in convex set, the Poincaré lemma guaranties that, there is a $2p-$form $\theta_A$ such that: $\mathrm{d}\theta_A = A\land \text{d}A.$
My question is how to explicitly compute the components of $\theta_A$. For example, for the case of $A$ being a 1-form: $\frac{1}{2!} \frac{\partial \theta_{\mu\nu}}{\partial x^\sigma} = \frac{1}{3!} \epsilon^{\alpha\beta\gamma}_{\mu\nu\sigma} A_{\alpha}\frac{\partial A_{\gamma}}{\partial x^\beta}$, but, for me, it is not clear how to proceed from this point.