 Solution

For a real skew-symmetric matrix, $P^T=-P$. A scalar quadratic form satisfies $y^{T}Py=(y^{T}Py{)}^T=y^T{P}^{T}y=-y^{T}Py$, so it is zero. ThereforeThe finite-dimensional linear system has the global solution $e^{tP}{y}_0$. Equivalently, $(e^{tP}{)}^T{e}^{tP}=e^{-tP}{e}^{tP}=I$, so its exact flow is an orthogonal matrix and preserves the Euclidean norm.