More interpretations:

On (pseudo-) Riemannian manifolds: the numerator of sectional curvature $-\langle R(X,Y)X,Y\rangle$ is a symmetric bilinear form on the space of skew-symmetric bivectors. Skew symmetric bivectors describe measured 2-planes in the tangent space. Curvature in the form of $\langle R(X,Y)Z,W\rangle$ can be recomputed by polarization from this.

Another, more general point of view: On the (orthonormal) frame bundle, curvature is a 2-form with values in the Lie algebra of the structure group: $\Omega=d\omega+\omega\wedge\omega$ for matrix valued forms.

More interpretations:

On (pseudo-) Riemannian manifolds: the numerator of sectional curvature $-\langle R(X,Y)X,Y\rangle$ is a symmetric bilinear form on the space of skew-symmetric bivectors. Skew symmetric bivectors describe measured 2-planes in the tangent space. Curvature in the form of $\langle R(X,Y)Z,W\rangle$ can be recomputed by polarization from this.

Another, more general point of view: On the (orthonormal) frame bundle, curvature is a 2-form with values in the Lie algebra of the structure group: $\Omega=d\omega+\omega\wedge\omega$ for matrix valued forms. This ties in well with the fact that curvature is the obstruction against integrability of the horizontal subbundle of $TTM$.