I want to find out if differentiable manifold of matrices $M=\{A_{(3\times3)}(\mathbb{R}): A^TA=16E\}\subset\mathbb{R}^{3\times3}=\mathbb{R}^9$ is orientable. It is only worth proving that orthogonal matrices form (or do not form) an orientable differentiable manifold.
I have tried to show that if there is a loop $\gamma(t)$, then normal vector $\vec{n}(t)$ to $M$ constructed from $\gamma(t)$ remains its orientation: $\forall t_0\;\vec{n}(t_0)=\vec{n}(t_0+1)$, but unfortunately got no results.