  We study the structure of the algebra of polynomial invariants for the usual
conjugation action of the complex special, SO_n, and general, O_n, orthogonal
group on the space of traceless n by n complex matrices. (Note that these two
algebras coincide if n is odd.) Minimal generating sets of these algebras are
known for n less than 5. We construct one for n=5. We also construct a Hironaka
decomposition in the case n=3 and a new (more economical) such decomposition
for n=4. A simple presentation (with just one syzygy) is obtained for the
algebra in the case n=3.
