I have the following exercise:
Show that $SO(n)$ is connected, using the following outline: For the case $n = 1$, there is nothing to show, since a $1\times 1$ matrix with determinant one must be $[1]$. Assume, then, that $n \geq 2$. Let $e_1$ denote the unit vector with entries $(1,0,\ldots, 0)$ in $\mathbb R^n$. Given any unit vector $v\in\mathbb R^n$, show that there exists a continuous path $R(t)$ in $SO(n)$ with $R(O) = I$ and $R(1)v = e_1$. Now, show that any element $R$ of $SO(n)$ can be connected to a block-diagonal matrix of the form \begin{pmatrix} 1 &\\&R_1 \end{pmatrix} with $R_1\in SO(n- 1)$ and proceed by induction.
I have troubles only with the first part, i.e., Given any unit vector $v\in\mathbb R^n$, show that there exists a continuous path $R(t)$ in $SO(n)$ with $R(O) = I$ and $R(1)v = e_1$. Please help me, thank you.