Orthogonal bundles and skew-Hamiltonian matrices

Using properties of skew-Hamiltonian matrices and classic connectedness results, we prove that the moduli space $M_{ort}^0(r,n)$ of stable rank $r$ orthogonal vector bundles on ${\mathbb{P}}^2$, with Chern classes $(c_1,c_2)=(0,n)$, and trivial splitting on the general line, is smooth irreducible of dimension $(r-2)n-\left(\genfrac{}{}{0px}{}{r}{2}\right)$ for specific values of $r$ and $n$.