Electromagnetic waves in tensor notation

I’m trying to derive the wave equations for the electric and magnetic fields in covariant (tensor) formulation.
Starting with Gauss-Ampere law,
$$
partial_alpha F^{alphabeta}=frac{4pi}{c}J^beta
$$
and Gauss–Faraday law
$$
partial_{alpha}(tfrac{1}{2}epsilon^{alphabetagammadelta}F_{gammadelta}) = 0
$$
I would like to derive the values of $square vec{E}$ and $squarevec{B}$, or in general get to $partial^{nu} partial_{nu} F^{alphabeta}$. Is there an elegant way to do this? I did try using summation in each of the equations, but got quite confused with all the indices.