I understand that if I define electric field to be $E=E_i dx^i$, magnetic field to be $B=B_1 dx^2 \wedge dx^3 + B_2 dx^3 \wedge dx^1 + B_3 dx^1 \wedge dx^2 $, and field strength to be $F= dx^0 \wedge E + B$, I would get the two homogenuous Maxwell equations from $dF=0$. The last equation is a nontrivial equation.
However I've read somewhere else that if I define the vector potential to be the one form $A=A_\mu dx^{\mu}$, and the field strength to be $F=dA$ I would get the two homogenuous Maxwell equations.
My questions:
First, with this second definition the equation $dF=0$ is trivially true since $d^2=0$ and I don't understand how this would give me anything non-trivial.
Secondly, from the second definition if I write $F$ in components I would have [*]: $$ F=dA=\partial_{\beta}A_{\mu} dx^{\beta} \wedge dx^{\mu}\\ = \partial_0 A_i dx^0 \wedge dx^i + \partial_i A_0 dx^i \wedge dx^0 + \partial_j A_i dx^j \wedge dx^i\\ = \partial_0 A_i dx^0 \wedge dx^i + x^0 \wedge E + B $$ which has the first term extra compared to the first definition, and I don't have any reason that this term is zero. So, what am I missing here? Which of the the two above approaches are correct?
[*] I use Greek letter super/subscripts for 4 space-time components and small english letters for 3 space componenets.