I am reading Petersen's "Riemannian Geometry" (3rd edition) and have come across some content which I do not understand. The problems lies in page 345 of the book which is seen here below.
Here $R_{X,Y}$ is the map $R_{X,Y}Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z$.
The second last line of the text says that there is a metric on $\mathfrak{so}(T_pM)$ induced by the metric on $\Lambda^2T_pM$ but what is this induced metric on $\mathfrak{so}(T_pM)$?
In the second line of the equation how should I make sense of $g(\mathfrak{R}(X\wedge Y),\Xi_{\alpha})$? From my understanding $\mathfrak{R}$ is a map from and into $\Lambda^2T_pM$ while $\Xi_{\alpha}$ is a map from and into $T_pM$.