This is question 4.5.11 in Artin's Algebra, first edition.
Let $v$ be a vector of unit length, and let $P$ be the plane in $\mathbb{R}^{3}$ orthogonal to $v$. Describe a bijective correspondence between points on the unit circle in $P$ and matrices $\mathit{P} \in SO_{3}(\mathbb{R})$ [I find $\mathit{P}$ a weird choice of name of matrix, I will use $\mathbf{P}$ instead] whose first column is $v$
I know the following: If $v$ has unit length and is orthogonal to $P$, then $\mathbf{P}\mathbf{P}^{T}=I$. Likewise, because $P$ is the unit circle, we can think of any point in $P$ as a rotation in the plane orthogonal to $v$. So that, if we focus only on that plane, we get a matrix $\mathbf{P}' \in SO_{2}(\mathbb{R})$ with the usual properties such as $\mathbf{P}'(\mathbf{P}')^{T}=I_{2}$. I made a figure of my geometric understanding of the question:
The grey disc is the unit circle orthogonal to $v$.
Is my understanding correct? How can we construct such bijection?