Given two linearly independent unit vectors $u, v \in \mathbb{R}^{n,1}$, the matrix $$ P(u, v) = I + 2 vu^{T}-\frac{(u+v)(u+v)^{T}}{1+u^Tv} $$ is the unique special orthogonal matrix that brings $u$ to coincide with $v$ and reduces to the identity when projected to the orthogonal complement of $\mathrm{span}(u,v)$. That is: $$P^T(u,v) P(u,v) = I$$ $$P(u,v)u = v$$ $$P(u,v)w = w\quad \forall w \in \mathbb{R}^{n,1}\ /\ u^Tw = 0 \wedge v^Tw = 0$$ (Note that for $n > 3$ not all special orthogonal matrices can be expressed in this way.)
I've never encountered this parametrization before today. Does it have a name?