The $\mathfrak{so}(n)$ algebra has some relations between generators always indicated as $$\left[T_{ij}, T_{kl}\right] = \delta_{ik}T_{jl} - \delta_{jk}T_{li} - \delta_{jl} T_{ik} + \delta_{il}T_{jk}$$
now I would like to know how to indicate this in a vector rep, i.e. by objects intrinsically with one index. In particular, I am asking the explicit form of the structure constants $$\left[T_a, T_b\right] = F_{ab}^{\phantom{ab}c}T_c\, .$$