General Formulas of the Structure Constants in the $\mathfrak{su}(N)$ Lie Algebra

We provide the analytic expressions of the totally symmetric and anti-symmetric structure constants in the $\mathfrak{su}(N)$ Lie algebra. The derivation is based on a relation linking the index of a generator to the indexes of its non-null elements. The closed formulas obtained to compute the values of the structure constants are simple expressions involving those indexes and can be analytically evaluated without any need of the expression of the generators. We hope that these expressions can be widely used for analytical and computational interest in Physics.