Perhaps this is a bit naïve, but I was wondering if it possible to (at least formally) represent Bundle Gerbes as Characteristic Classes. Disclaimer: My understanding of Bundle Gerbes is limited to this paper of Hitchin so perhaps I'm not thinking of this correctly. Just for reference, a Bundle Gerbe is defined by specifying an open cover $\{U_i\}$ of a manifold $M$ that has associated to it maps $g_{ijk} : U_i \cap U_j \cap U_k \rightarrow S^1$ that satisfy certain cocycle-like conditions, $g_{jkl} g_{ikl}^{-1} g_{ijl} g^{-1}_{ijk}$. One can define connective structures with $3$-form curvatures $H$ on Bundle Gerbes that define principle circle bundles on the Loop Space of $M$ (See Hitchin, Page 4). These connective structures are classified by their curvatures, $[H / 2\pi] \in H^3(M,\mathbb{Z})$ just like the curvature $2$-form of a line bundle generates the first Chern Class. Explicitly, my question is the following:

Can we expand the definition of a bundle gerbe on a manifold $M$ to an arbitrary compact, finite-dimensional Lie Group $G$ by considering a Bundle Gerbe to instead be the set of maps $g_{ijk} : U_i \cap U_j \cap U_k \rightarrow G$? If $\dim G = n$, will $H^{n}(M,\mathbb{Z})$ classify the Principle $G$-bundles on $\Omega M$?

Again, my understand of gerbes is quite insufficient so perhaps this is "obvious" in some other literature. If this is the case, could you please cite a reference?

Thanks!

PS: I'm not sure if the compactness is truly necessary, I just added it with the hope that its more likely in the compact case