  This paper studies differential graded modules and representations up to
homotopy of Lie $n$-algebroids, for general $n\in\mathbb{N}$. The adjoint and
coadjoint modules are described, and the corresponding split versions of the
adjoint and coadjoint representations up to homotopy are explained. In
particular, the case of Lie 2-algebroids is analysed in detail. The
compatibility of a Poisson bracket with the homological vector field of a Lie
$n$-algebroid is shown to be equivalent to a morphism from the coadjoint module
to the adjoint module, leading to an alternative characterisation of
non-degeneracy of higher Poisson structures. Moreover, the Weil algebra of a
Lie $n$-algebroid is computed explicitly in terms of splittings, and
representations up to homotopy of Lie $n$-algebroids are used to encode
decomposed VB-Lie $n$-algebroid structures on double vector bundles.
