I have read somewhere that the general form of a metric affine connection whose Riemann curvature is zero is given by $$\Gamma^{i}_{ij}=A^{i}_{\alpha}\partial_{j}A^{\alpha}_{k}$$ where $A^{i}_{\alpha}$ are elements of some invertible matrix. This form comes from a general solution of the partial differential equations

$$R^{i}_{jkl}\equiv 2(\partial_{j}\Gamma^{i}_{kl}-\Gamma^{i}_{kp}\Gamma^{p}_{jl})_{[jk]}=0.$$ How does the above form of the connection coefficients is a general solution to the system of pde's? Substitution of the expression is obviously a proof, but where does that form come from?