  We provide a closed formula for the degree of $\text{SO}(n)$ over an
algebraically closed field of characteristic zero. In addition, we describe
symbolic and numerical techniques which can also be used to compute the degree
of $\text{SO}(n)$ for small values of $n$. As an application of our results, we
give a formula for the number of critical points of a low-rank semidefinite
programming optimization problem. Finally, we provide some evidence for a
conjecture regarding the real locus of $\text{SO}(n)$.
