 Solution

Write $n=2m-1$. On ${\mathbb{R}}^{2m}$ define the orthogonal linear mapIt satisfies $J^T=-J$ and $J^{T}J=I$. Hence $⟨x,Jx⟩=0$ and $\Vert Jx\Vert =\Vert x\Vert$. Restricting to the unit sphere givesThis continuous unit tangent field is available exactly because the ambient dimension is even.