It's not quite clear what a "description" should consist of; for example, in higher ranks one can't draw pictures. But a certain amount of information about the geometry of weight polytopes has been developed in recent papers by Apoorva Khare, probably in more or less generality than you want. (His main interest is in infinite dimensional highest weight representations, but he also considers finite dimensional ones.)

Khare's papers are posted on arXiv, and at least one co-authored by him has been published: see the preprint. But it's probably most useful to contact him directly at Stanford. While the fundamental representations are obviously easier to study in many respects, it's essential to formulate your question as precisely as possible.

It's not quite clear what a "description" should consist of; for example, in higher ranks one can't draw pictures. But a certain amount of information about the geometry of weight polytopes has been developed in recent papers by Apoorva Khare, probably in more or less generality than you want. (His main interest is in infinite dimensional highest weight representations, but he also considers finite dimensional ones.)

Khare's papers are posted on arXiv, and at least one co-authored by him has been published: see the preprint. But it's probably most useful to contact him directly at Stanford. While the fundamental representations are obviously easier to study in many respects, it's essential to formulate your question as precisely as possible.