In order to derive the inhomogeneous Maxwell equations, you do not need to use the antisymmetry of the field strength tensor. They follow from the equations of motion for $A_\mu$. If you start with $$ \mathscr{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-e J^\mu A_\mu\;,$$ the Euler-Lagrange equations give you $$ \partial_\mu F^{\mu\nu}=e J^\nu$$ up to a total derivative term that can be gauged or argued away. If you relabel $\mu\leftrightarrow\nu$, you get $$ \partial_\nu F^{\nu\mu}=e J^\mu\;.$$ It might very well be that our conventions differ because I do not work with these $\mu_0$ oddities. That is, all the above assumes natural units, in which $c=1$. In any case, you do not get a sign when renaming the summation indices.
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I mistakenly approved an edit which had one very good point, but several other not so good things.
user178876