  We study the representation theory of finite-dimensional $\omega$-Lie
algebras over the complex field. We derive an $\omega$-Lie version of the
classical Lie's theorem, i.e., any finite-dimensional irreducible module of a
soluble $\omega$-Lie algebra is one-dimensional. We also prove that
indecomposable modules of some three-dimensional $\omega$-Lie algebras could be
parametrized by the complex field and nilpotent matrices. We introduce the
notion of a tailed derivation of a nonassociative algebra $g$ and prove that if
$g$ is a Lie algebra, then there exists a one-to-one correspondence between
tailed derivations of $g$ and one-dimensional $\omega$-extensions of $g$.
