The so called Gell-Mann formula expresses the Lie algebra elements in terms of the corresponding Inonu-Wigner contracted ones. In the case of sl(n, R) and su(n) algebras contracted w.r.t. so(n) subalgebras, the Gell-Mann formula is generally not valid, and applies only in the cases of some algebra representations. A generalization of the Gell-Mann formula for sl(5,R) and su(5) algebras, that is valid for all representations, is obtained in a group manifold framework of the SO(5) and/or Spin(5) group.