  In this paper we explore the structure of certain generalized isometries of
the special orthogonal group $SO(n)$ which are transformations that leave any
member of a large class of generalized distance measures invariant. This gives
us a far-reaching generalization of the former structural result of T. Abe, S.
Akiyama and O. Hatori concerning $c$-spectral isometries of the special
orthogonal group.
