There are different ways of defining and thereafter calculating the Chern classes. Right now I'm studying from the lecture notes which introduce the first Chern class through the classifying spaces as follows:
The classifying bundle for $\mathbb{U}(1)$ is $\mathbb{S}^{\infty} \rightarrow \mathbb{CP}^{\infty}$. So the set of complex line bundles over $B$, which are $\mathbb{U}(1)$-bundles, is in the bijective correspondence with the maps from $B$ to $\mathbb{CP}^{\infty}$ taking up to homotopy. So for any bundle there exists a map induced on the cohomology rings from $\mathbb{Z}[t]$ to $H^{\bullet}(B)$. And the first Chern class is defined as the image of the generator $t$ in the latter map.
I have a canonical line bundle over $\mathbb{CP}^n$ and I want to calculate the first Chern class using classifying space theory. Do I have to follow the explicit construction of the map between $\mathbb{CP}^n$ and $\mathbb{CP}^{\infty}$ for this line bundle, because I couldn't do it, or there is another way using classifying spaces?