Interpretation of Wikipedia formula relating curvature form and curvature tensor

Question

According to Wikipedia (see here), the curvature $2$-form $$ \Omega_{j}^{i} = d\omega_{j}^{i}+\omega_{k}^{i}\wedge\omega_{j}^{k}\, , $$ is related to the Riemann curvature endomorphism by $$ R(X,Y)Z = \nabla_{X}\nabla_{Y}Z + \nabla_{Y}\nabla_{X}Z + \nabla_{[X,Y]}Z = \Omega(X\wedge Y)Z\,. $$

I am confused because $\Omega(X\wedge Y)$ is a function (is not it?), and so it seems that one could conclude that $R(X,Y)Z$ is parallel to $Z$, which is clearly false. So how should I interpret this equation?

Answer 1

$\Omega(X\wedge Y)$ is an endomorphism of the tangent space $T_pM$ (I take $X,Y\in T_pM$; there is a similar interpretation if $X$ and $Y$ are vector fields). It is defined by $$ \bigl(\Omega(X\wedge Y)Z\bigr)^i = \bigl(d\omega^i_j(X,Y) + \omega_k^i(X)\omega_j^k(Y) - \omega_k^i(Y)\omega_j^k(X)\bigr)Z^k . $$ Stated more like the Wikipedia entry, $\Omega_j^i$ is an $n$-by-$n$ matrix of two-forms, and $\Omega(X\wedge Y)$ denotes the $n$-by-$n$ matrix of real numbers obtained by evaluating each entry at $X\wedge Y$. This acts on $Z$ by regarding $Z$ as a column vector.

Answer 2

This is just a restatement of Jeffery Case answer. If I understand correctly, $\Omega(X\wedge Y)$ is not a function. It is a $n\times n$ matrix of functions and we have $$\Omega(Y\wedge Y)Z=\begin{bmatrix} \Omega^1_1&\cdots &\Omega^1_{n}\\ \vdots& \ddots&\vdots\\ \Omega^{n}_1&\cdots&\Omega^{n}_{n} \end{bmatrix}\begin{bmatrix}Z^1\\\vdots\\ Z^n\end{bmatrix}.$$