Could any one give me hint how to show $SO(n)$ is connected? I understand that it is closed subset, I can prove $O(n)$ is not connected. Edit: suddenly got some idea, any matrix from $SO(n)$ can be written in this from, where first and second row are like below $\left(\begin{array}{cc} \cos\theta & \sin\theta\\ -\sin\theta & \cos\theta\end{array}\right).$
we can make $f(t)= tA+(1-t)I$ this is a path between $A$ and $I$. can this idea be rigorised?