# Construct a trivializing cover of universal principal $G$-bundle $EG \to BG$

In Hatcher's book on algebraic topology (p 89) the universal bundle $$EG$$ ($$G$$ discrete group) carries structure of a $$\triangle$$-complex whose $$n$$-simplices are the ordered $$(n + 1)$$ tuples $$[g_0, ... ,g_n]$$ of elements of $$G$$. As quotient space the classifing space $$BG=EG/G$$ inherits structure $$\triangle$$-complex where a $$n$$-simplex $$BG$$ can be written uniquely in the form $$[g_1 \vert g_2\vert ... \vert g_n]:= G \cdot [e_G, g_1, g_1g_2,..., g_1,.., g_n]$$.

My question is if there is a natural way to choose a trivializing cover $$\{U_i\}_{i \in I}$$ of the universal principal $$G$$-bundle $$EG \to BG$$, ie a family of open subsets of $$BG$$ such that $$EG \vert \_{U_i} \cong U_i \times G$$ and what are the the associated transition functions $$h_{ij}: U_i \cap U_j \to G$$ used for patching functions $$U_i \cap U_j \times G \to U_i \cap U_j \times G, (u,g) \mapsto (u, h_{ij}(u)g)$$.

The question is closely connected to a problem appearing in a construction here