What Conditions Make a Matrix the Hessian of a Function?

[quasar987]

Apparently it is a well-known fact that if $G(x)=(G_{ij}(x_1,\ldots,x_n))$ is a smooth nxn matrix-valued function such that $G_{ij,k}=G_{ik,j}$ for all i,j,k, then there exists a smooth function g such that Hess(g)=G; i.e. $g_{,ij}=G_{ij}$. ($f_{,k}$ denotes partial differentiation with respect to the kth variable.)

I believe I can construct the solution explicitly in the n=2 case, but I'm not sure how to generalize my argument. Is there an argument to be made about the existence of a solution to this overdetermined system of PDE? Thx!