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

Consider a non zero vector $v$ at the tangent space of identity matrix. Denote the corresponding matrix multiplication by $\phi_A$. Define the vector field $F$ by $F(A)=(\phi_A)_*(v)$. Where $\phi_*$ is the derivative of $\phi$, and $v$ is a tangent vector at identity with a fixed direction.
So $$A = \begin{pmatrix} \cos (\pi/2) & -\sin (\pi/2) & 0 \\ \sin(\pi/2) & \cos(\pi/2) & 0\\ 0&0&1 \end{pmatrix}$$ is homotopic to identity map.
Then how shall I proceed....?
Thank you~~~

Are you familiar with the theory of Lie Groups? You can just take any non-zero vector at the identity and translate it everywhere, generating a non-vanishing smooth vector field on $O(3)$. From here it's easy with Poincare-Hopf.
