Given a metric, for example

$$
ds^2 = -A(r)dt^2 + B(r)dr^2 + C(r)d\theta^2 + D(r) d\phi^2,
$$
and assuming that the fields go as
$$
\textbf{E} = E(r)\hat{r} \quad \text{and} \quad \textbf{B}=0,
$$
how does one compute the EM field tensor from this? I keep trying to express the "standard" field tensor,
$$
F^{\mu \nu }={\begin{bmatrix}0&-E\_{x}&-E\_{y}&-E\_{z}\\E\_{x}&0&-B\_{z}&B\_{y}\\E\_{y}&B\_{z}&0&-B\_{x}\\E\_{z}&-B\_{y}&B\_{x}&0\end{bmatrix}}
$$
in spherical coordinates but find myself just ignoring the metric. I've also tried going back to the definition,
$$
F\_{\mu\nu} = \partial\_\mu A\_\nu - \partial\_\nu A\_\mu
$$
but I am unsure how to express the four-derivatives and four-potentials. It seems like I'm missing something fairly fundamental. Any help would be appreciated.