I want to understand certain properties of general curves $\textrm{Ad}_{e^{t X}} = e^{t ad_X}$ of adjoint-representation matrices of $SU(N)$. For this purpose, I would like to have an explicit closed-form expression for all structure constants $f_{abc}$ of $\mathfrak{su}(N)$ for arbitrary $N$ in the form $[T_a, T_b]=f_{abc} T_c$. Ideally, I would like to have them for the usual choice of basis $\{T_a\}$ with the convention that all $T_a$ are traceless, anti-Hermitian and trace-orthogonal, with standard Dynkin index $ tr(T_a T_b) = - \frac{1}{2} \delta_{ab} $, in which $f_{abc}$ is totally antisymmetric.
Is there a reference where such expression can be found? Otherwise, I am more or less familiar with the machinery of the Cartan classification. What would be the best way to attack the problem?