Exponential of a skew-Hermitian matrix ($A^{†}=-A⟹(e^A{)}^{†}{e}^A=I$)

If $A$ is Skew-Hermitian, then $-iA$ is a Hermitian matrix. The finite-dimensional spectral theorem gives $A=Q\mathrm{diag}(i{\theta }_j)Q^{†}$, with real ${\theta }_j$ and a unitary matrix $Q$. Since the matrix exponential respects similarity, $e^A=Q\mathrm{diag}(e^{i{\theta }_j})Q^{†}$. Each diagonal entry has modulus one, so $e^A$ is a unitary matrix. Thus a skew-Hermitian matrix generates norm-preserving linear evolution.