I'm interested in writing down explicitly the classifying map for a given real vector bundle $\pi\colon E \to B$ of rank, say $k$. Take $B$ to be a compact manifold (I'm interested in this case) so we can assume that the transition functions lies in $O(k)$. As a model for the universal $O(k)$ principal bundle we take the usual bundle $V_k(\mathbb{R}^{\infty})\to Gr_k(\mathbb{R}^{\infty})$ (both of them are build via the respective colimits).
First step is to consider the associated $O(k)$ bundle $\tilde{E}$ defined as (following tom Dieck's notation: page 340) $\coprod_{b\in B} \text{Iso}(\mathbb{R}^k,E_b)$ where $\text{Iso}(\mathbb{R}^k,E_b)$ is the set of linear isomorphism between a fibre and the canonical vector space. Or analogously one can take the frame bundle associated to it (see this answer for example).
Problem. if we take tom Dieck definition and/or the Frame Bundle, I don't see a canonical way to map the bundle inside $EO(k)$. Morally it's easy because (in the frame bundle case for example) every point is a $k$-frame and $EO(k)$ can be seen as the set of $k$-frames. But if we want to be precise, the points in the frame bundle are frames in the vector space $E_b$, for any $b\in B$, the target space contains instead the $k$-frames in $\mathbb{R}^{\infty}$. So how can we write down explicitly a map sending the former in the latter?
The only thing that came to my mind was to choosing a basis basis/isomorphism $E_b \to \mathbb{R}^k$ for each fibre. I'm unsure about that though. Can someone provides some references or clarification/comments about this?
There's no hope of getting a canonical point-set classifying map, since even the classifying space is only unique up to homotopy equivalence. As far as writing anything down explicitly, for the tangent bundle you can do the following. Pick an embedding of your manifold $X$ into $\mathbb{R}^n$ for some large $n$ (you really do need to make a choice approximately this noncanonical). Then you can associate to each point $x \in X$ the subspace of $\mathbb{R}^n$ (canonically identified with the tangent space at each of its points) spanned by the tangent space $T_x(X)$. This gives a map from $X$ to a Grassmannian, and composing with the map from this Grassmannian to the corresponding infinite Grassmannian gives a classifying map for the tangent bundle.
For a general bundle you need to pick an embedding of the bundle into a trivial bundle.
Conceptually, what you'd like to do is the following. The classifying space for vector bundles is intuitively "the space of all vector spaces." Then you'd like to assign to every $x \in X$ the tangent space $T_x(X)$, as a point in the "space of all vector spaces." But it's pretty unclear what this means. One way to write down a point-set construction of this is to instead take the "space of all subspaces of $\mathbb{R}^{\infty}$," and this is what the Grassmannian construction of the classifying space does. Then to write down maps into this you need to write down embeddings into $\mathbb{R}^{\infty}$.