Write electromagnetic field tensor in terms of four-vector potential

The Bianchi identity $\mathrm{d}F~=~0$ together with Poincare lemma guarantee the local existence of $A$ in contractible regions of spacetime. See also this related Phys.SE post.

One way to write the homogenous Maxwell's equations is with the Levi-Civita symbol $\epsilon$: $$\epsilon^{\alpha\beta\mu\nu} \partial_\beta F_{\mu\nu} = 0$$

Solution to this is obviously (with arbitrary potential $A$): $$F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$$

It is easy to verify using the antisymmetry of $\epsilon^{\alpha\beta\mu\nu}$ upon swapping any 2 indexes, together with $\partial_\mu\partial_\nu = \partial_\nu\partial_\mu$.