Let $\mathcal{G}$ be a generalized matrix algebra. We show that under certain conditions, each generalized Lie $n$-derivation associated with a linear map on $\mathcal{G}$ is a sum of a generalized derivation and a central map vanishing on all $(n-1)$-th commutators and is also a sum of a generalized inner derivation and a Lie $n$-derivation. As an application, generalized Lie $n$-derivations on von Neumann algebras are characterized.