All you need to know is how to build a rank $n$ vector bundle from a principal $GL(n)$-bundle. The idea there is to view a local section of the principal bundle as a local frame of a vector bundle, and the transition functions for the principal bundle as change of frame maps for the vector bundle.

Then given any principal $G$-bundle and a representation $G \rightarrow GL(V)$, there is a naturally defined principal $GL(V)$-bundle whose transition functions are defined by composing the transition functions of the original bundle with the representation.