<abstract><p>In this paper, we introduce the notion of Lie $ n $-centralizers. We then give a description of Lie $ n $-centralizers on a generalized matrix algebra and present the necessary and sufficient conditions for a Lie $ n $-centralizer to be proper. As applications, we determine generalized Lie $ n $-derivations on a generalized matrix algebra and Lie $ n $-centralizers of some operator algebras.</p></abstract>