How to find a local parameter of the tangent space at any point of a unit $n$-sphere?

Question

I need to find an analytic form of the local parametrisation of the tangent space at any point of a unit $n$-sphere.

While I know the same for $1$-sphere, $2$-sphere, how can I derive the general form?

Answer 1

For any point $\mathbf{p}\in\mathbb{R}^n$ on the unit $(n\textrm{-}1)$-sphere, the tangent space is orthogonal to the vector equal to the point $\mathbf{p}$ itself. So, the tangent space at the point $\mathbf{p}$ is given by its orthogonal subspace, namely, $\mathbf{I}-\mathbf{p}\mathbf{p}^T$.