I wish to derive the Lie algebra $so(4)$ of the generators $J_{jk}=-J_{kj}$ of ${\rm SO(4)}$ as $$[J_{jk},J_{lm}]=\delta_{jl}J_{km}+\delta_{km}J_{kl}-\delta_{jm}J_{kl}-\delta_{kl}J_{jm}.\tag{1}$$ However, to derive this, one requires the expression $$(J_{jk})_{ab}=-i(\delta_{ja}\delta_{kb}-\delta_{jb}\delta_{ka})\tag{2}$$ How is (2) obtained? I know that for a Lie group element $g=e^{-\frac{i}{2}\theta_{jk}J_{jk}}$ and the generators are defined by $$J_{jk}=-i\frac{\partial g}{\partial\theta_{jk}}|_{\{\theta_{kj}=0\}}.$$
Any idea how to derive Eq.(2)?
This is but the defining N×N matrix representation of so(N), consisting of N(N-1)/2 imaginary antisymmetric matrices (in physics, so they are Hermitean!), labelled by the location-reminder antisymmetrized integer pair indices, of course also N(N-1)/2 in number; (2) is manifestly a complete basis for them.
Both (1) and (2) hold for so(N), not just so(4), since exponentials of real antisymmetric matrices are real orthogonal matrices, which preserve the real dot product of N-vectors.
In particular, you are already familiar with the SO(3) generators, $J_{jk}$, alias $J^i\equiv \epsilon^{ijk} J_{jk}$, and the Lie algebra expressed in this language.