  For the nonstandard $q$-deformed algebras $U_q(so_n)$, defined recently in
terms of trilinear relations for generating elements, most general finite
dimensional irreducible representations directly corresponding to those of
nondeformed algebras $so(n)$ (i.e., characterized by the same sets of only
integers or only half-integers as in highest weights of the latter) are given
explicitly in a $q$-analogue of Gel'fand-Tsetlin basis. Detailed proof, for $q$
not equal to a root of unity, that representation operators indeed satisfy
relevant (trilinear) relations and define finite dimensional irreducible
representations is presented. The results show perfect suitability of the
Gel'fand-Tsetlin formalism concerning (nonstandard) $q$-deformation of $so(n)$.
