Any skew-symmetric tensor $ F_{\alpha\beta} $ which is the curl of a 4-vector $A_\mu$, that is each tensor having the form $ F_{\alpha\beta} = \partial_\alpha A_\beta - \partial_\beta A_\alpha $, will satisfy the relation $$ \partial_\alpha F_{\beta\gamma} + \partial_\beta F_{\gamma\alpha} + \partial_\gamma F_{\alpha\beta} = 0. $$
That is very easy to see. But I have some heuristic reasons (related to the electromagnetic tensor) to think that the converse is also true.
Question: How can you prove that if $ \partial_\alpha F_{\beta\gamma} + \partial_\beta F_{\gamma\alpha} + \partial_\gamma F_{\alpha\beta} = 0 $ then $F_{\alpha\beta}$ is necessarily the curl of a 4-vector $A_\mu$?