Alternate proposed differential form of Maxwell action

Question

The electromagnetic action is defined by the wedge product of the electromagnetic tensor F and its hodge star *F

What would be the explicit form of the action if this would be substituted with the wedge product of F and F instead?

I obtain an identically zero answer but I am doubting it very much.

Answer 1

OP's result is correct: $F\wedge F$ is a locally exact 4-form, which means it doesn't contribute to Euler-Lagrange (EL) equations, cf. e.g. this Phys.SE post. The term $F\wedge F$ is known as a topological term, see e.g. this Phys.SE post.