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. Now use the construction above to construct the vector bundle from the principal $GL(V)$-bundle.