Riemannian curvature space

Definition

Let $V$ be a real vector space with a Euclidean inner product. The Riemannian curvature space of $V$ is the space of $(1,3)$-tensors on $V$ of the following description:

$R$ is in the Riemannian curvature space if $R(X,Y,Z)$ is alternating in $X$ and $Y$, and further, for any $X,Y$, the map $(Z,W)\mapsto g(R(X,Y,Z),W)$ is alternating in $Z$ and $W$.

By the canonical identification of $(1,3)$-tensors with $(0,4)$-tensors, the Riemannian curvature space can be identified with the symmetric square of the exterior square of the vector space.