Given a principal $G$-bundle $(P, M, G)$ ($G$ is a topological group), for any vector space $V$ and a representation $\rho: G \rightarrow \operatorname{GL}(V)$, I can build an associated vector bundle $(P \times_{\rho}V, M, V)$. On the other hand, given a vector bundle $(E, M, V)$, the frame bundle $F(E)$ associated to vector bundle E is a principal bundle.
My question is: If I take the $G = \operatorname{GL}(n, \mathbb{K})$ as groups ($\mathbb{K}$ is a field), is there an equivalence between the categories $\textbf{Bun}_{G}(M)$ of principal $G$-bundles over $M$ and $\textbf{Vec}_{\mathbb{K}}^{n}(M)$ of $n$-dimensional vector bundles over $M$?
Thanks in advance.