Recall that the orthogonal group \(O(n)\) is
\[ O(n)=\{M\in \mathcal {M}_{n\times n}\mid MM^{\top }=I\} \]
where \(I\) is the identity matrix. Identifying the set \(\mathcal {M}_{n\times n}\) of all real \(n\times n\) matrices with \(\mathbb {R}^{n^2}\), show that \(O(n)\) is a submanifold of \(\mathcal {M}_{n\times n}\) and compute the tangent space to \(O(n)\) at \(I\) as a linear subspace of \(T_I\mathcal {M}_{n\times n}\), which is identified with \(\mathcal {M}_{n\times n}\).