The conditions which determine whether or not a matrix is special orthogonal are polynomial in the matrix entries and thus give an explicit description of SO(n) as an embedded algebraic variety. We give a formula for the degree of this variety for any n which is interpretable as counting non-intersecting lattice paths. This degree also contributes to the degree of low-rank semidefinite programming. We explain how to verify the formula explicitly using numerical algebraic geometry (for $n\leq7$) and how numerical computations aid in the study of the real locus of this variety.