From a bank of masters exams:
Say the position of a particle moving in $\mathbb{R}^n$ is given by a smooth vector-valued function $\vec{x}(t)$. Suppose that $\vec{x}(t)$ satisfies a differential equation, $ \frac{d\vec{x}}{dt} = A(t)\vec{x},$ where $A(t)$ is a real anti-symmetric matrix depending smoothly on $t$. Show that this particle moves on a sphere, that is, $||\vec{x}(t)||$ is constant.
By the spectral theorem, $A$ is normal and therefore has a complete basis of eigenvectors in $\mathbb{C}^n$. I am familiar with the "standard" method of solving for matrix exponentials, i.e. finding the eigenvalues and eigenvectors of $A$, and then using linear combinations of $e^{\lambda t}\vec{x}$ as the solutions, but there is not a complete basis of eigenvectors in $\mathbb{R}$. Taking the matrix exponential $e^A$ doesn't seem to do anything.