In computing the topological $K$-theory of $\mathbb{R}P^2$ I had an idea to mimic the clutching construction: Recall that this says that for the open cover $S^k = D^k_+\cup D^k_-$ a rank $n$ vector bundle over $S^k$ is uniquely specified by the homotopy class of a map $f: S^{k-1} = D^k_+\cap D^k_- \to GL_n(\mathbb{R})$
For $\mathbb{R}P^2$, I thought about using the open cover $\mathbb{R}P^2 = U_0 \cup U_1 \cup U_2$ where $U_i = \{[x_0, x_1, x_2] \mid x_i \neq 0\} \cong \mathbb{R}^2$, hence contractible. Then one gets trivial vector bundles over these sets, and on the intersection a map $f: U_0 \cap U_1 \cap U_2\to GL_n(\mathbb{R})$. I think I can compute the homotopy type of this intersection, but it is the map the other way I have trouble with, as we now have to demand some kind of compatability also on $U_i \cap U_j$.
So the question is what kind of data do we need to reconstruct the bundle, and is this something we can reasonably compute with homotopy theoretic tools?
Note: I am not just interested in the $K$-theory of $\mathbb{R}P^2$, but in computing it in this manner (or for other types of spaces for which a construction like this is feasible).
Apologies for the vagueness of the question, and I appreciate any help and clarification!