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In the general case of the $\mathfrak{so}(n)$ free rigid body we give a listof integrals of motion, which generate the set of Mishchenko's integrals. Inthe case of $\mathfrak{so}(5)$ we prove that there are fifteen coordinate typeCartan subalgebras which on a regular adjoint orbit give fifteen Weyl grouporbits of equilibria. These coordinate type Cartan subalgebras are theanalogues of the three axes of equilibria for the classical rigid body on$\mathfrak{so}(3)$. The nonlinear stability and instability of these equilibriais analyzed. In addition to these equilibria there are ten other continuousfamilies of equilibria.

On the stability problem for the $\mathfrak{so}(5)$ free rigid body