For any discrete group $G$, the classifying space $BG$ is a $K(G, 1)$ and can be constructed as a simplicial set with $|G|^i$ $i$-simplices. Accordingly, elements of $H^i(BG, A)$ can be represented by simplicial $A$-valued $i$-cocycles, which are functions $$\omega: G^i\rightarrow A\;,$$ which leads to the algebraic definition of group cohomology.
Question: To which extent is it possible to apply the algebraic definition of group cohomology to continuous topological groups, such as $SO(3)$ or $U(2)$?
I believe the algebraic definition works at least for $G=U(1)=\mathbb{R}/\mathbb{Z}$. E.g., the generating element of $H(BU(1), \mathbb{Z})$ can be represented via the Bockstein homomorphism for the group extension $$\mathbb{Z}\rightarrow \mathbb{R}\rightarrow \mathbb{R}/\mathbb{Z}\;,$$ yielding a function $$\omega(a, b)=\widetilde{a}+\widetilde{b}-\widetilde{ab}\;,$$ with $a, b\in \mathbb{R}/\mathbb{Z}$, $\widetilde \cdot$ being some embedding of $\mathbb{R}/\mathbb{Z}$ into $\mathbb{R}$.
Is this also possible for other compact Lie groups and cohomology classes? If yes, what do the explicit cocycles looks like for some common examples, e.g., the first Chern class $c_1\in H^2(BU(2), \mathbb{Z})$?