Show that the Euler characteristic of $O[3]$ is zero.

Show that the Euler characteristic of $O[3]$ is zero.

Show that the Euler characteristic of $O[3]$ is zero.
So I am able to show that $$A = \begin{pmatrix} \cos (\pi/2) & -\sin
(\pi/2) & 0 \\ \sin(\pi/2) & \cos(\pi/2) & 0\\ 0&0&1 \end{pmatrix}$$ is
homotopic to identity map, and also degree is homotopic invariant. hope to
invoke Poincare-Hopf Index theorem. But I have trouble on constructing the
vector field with $O[3]$.
Thank you~~~