Let $R$ be the riemann curvature tensor, viewed as a $(0,4)$-tensor field. For $p \in M$, where we have a semi-riemannian manifold $(M,g)$, we can look at linearly independent vectors $u,v \in T_pM$ that spans a non-degenerate $2$-plane $\sigma$. According to notes from a course I am taking, the set $(u,v)$ of such pairs is dense in $T_pM \times T_pM$, and determines $R_p(u,v,v,u)$ by continuity. By "sets of such pairs", do they mean the set of all such pairs of vectors? That is, the set $$S := \{(u,v) \in T_pM \times T_pM: (u,v) \ \text{linearly independent and spans a 2-plane} \ \sigma\}.$$
Could someone point me to a proof of the statement about the set being dense, and why this determines $R_p(u,v,v,u)$ by continuity? The two note-sets I have does not prove this, but only mentions it.