is it like: $F^{0\nu} = F^{0i} = \partial^0 A^i - \partial^i A^0 = \partial_0 A^i + \partial_i A^0 = -E^i$? where I am using $(+, -, -, -)$ signature. where the final equality is our usual definition of the electric field. I think this is wrong because the indices get messed up in the penultimate equality...