Special orthogonal group as a submanifold ($T_{R}SO(n)=R\mathfrak{so}(n)$)

The map $A\mapsto A^{T}A$ from real matrices to symmetric matrices has derivative $H\mapsto R^{T}H+H^{T}R$ at $R\in O(n)$, which is surjective. Thus $SO(n)$ is a submanifold of dimension $n(n-1)/2$, and