Let's assume we start with Chern--Weil theory in the following form:

Given a manifold $M$ and a complex vector bundle $V$ over $M$, we can equip $V$ with a $\mathfrak g\mathfrak l_n(\mathbb C)$ connection and from this connection compute a closed differential $2k$-form (from the curvature of the connection) which thus determines an element $c_k(V)\in H^{2k}(M,\mathbb C)$ (by deRham theory). This value is independent of the connection chosen. If $f:N\to M$ is a smooth map, then $c_k(f^\ast V)=f^\ast c_k(V)$.

I've often heard that Chern--Weil theory gives cohomology classes $c_k\in H^{2k}(B\operatorname{GL}_n(\mathbb C),\mathbb C)$. However, if we take Chern--Weil theory to mean the above boxed summary, then this does not seem obvious to me unless we use the fact that we can choose $B\operatorname{GL}_n(\mathbb C)$ to be a direct limit of manifolds (namely Grassmannians).

My question is whether there is more abstract way of constructing the classes $c_k\in H^{2k}(B\operatorname{GL}_n(\mathbb C),\mathbb C)$ from Chern--Weil theory, using only the fact that $B\operatorname{GL}_n(\mathbb C)$ is the classifying space of complex vector bundles. I am fine with assuming that $B\operatorname{GL}_n(\mathbb C)$ is a direct limit of finite CW complexes, but of course, Chern--Weil theory does not obviously define Chern classes for vector bundles over CW complexes.