At the beginning of the paper "The curvature of 4-dimensional Einstein spaces", by Singer and Thorpe, the authors define the space $\mathcal{R}$ of curvature tensors of the vector space $V$ as the set of symmetric bilinear transformations on the space of bivectors $\Lambda^{2}(V).$
Few lines after this, they define the ''Bianchi map'' $b$ as an operator $b : \mathcal{R} \rightarrow \mathcal{R}$ in the following way: $$[b(R)](u_{1},u_{2})u_{3}=\sum_{\sigma \in S_{n}} R(u_{\sigma(1)},u_{\sigma(2)})u_{\sigma(3)},$$ but they do not explain the notation. If $R$ is a transformation $$R: \Lambda^{2}(V) \rightarrow \Lambda^{2}(V),$$ and $u_{1},u_{2}$ and $u_{3}$ are, I presume,vectors of $V,$ what is the meaning of $R(u_{\sigma(1)},u_{\sigma(2)})u_{\sigma(3)}$?
Also, they define immediately after this the ''Ricci contraction'' $r$ as an operator from $\mathcal{R}$ to the space of symmetric linear transformations of $V$ by means of: $$\langle r(R)(v),w \rangle=\mathrm{Tr}\{u \rightarrow R(v,u)w \}.$$
I see that this strongly resembles to the Ricci tensor that one usually meets in riemannian geometry, but I have a similar notational problem with this last definition.
If someone could possible clarify the notation and explain a little this way to look at the curvature tensor (or give me some references) I would be really grateful.