  We introduce a variant of the much-studied $Lie$ representation of the
symmetric group $S_n$, which we denote by $Lie_n^{(2)}.$ Our variant gives rise
to a decomposition of the regular representation as a sum of {exterior} powers
of modules $Lie_n^{(2)}.$ This is in contrast to the theorems of
Poincar\'e-Birkhoff-Witt and Thrall which decompose the regular representation
into a sum of symmetrised $Lie$ modules. We show that nearly every known
property of $Lie_n$ has a counterpart for the module $Lie_n^{(2)},$ suggesting
connections to the cohomology of configuration spaces via the character
formulas of Sundaram and Welker, to the Eulerian idempotents of Gerstenhaber
and Schack, and to the Hodge decomposition of the complex of injective words
arising from Hochschild homology, due to Hanlon and Hersh.
