Bijection Between Point-Hyperplane Anti-Flags of $V(n,2)$ and Non-Singular Points of $O^{+}(2n,2)$

We give a bijection between the point-hyperplane antiflags of $V(n,2)$ and the nonsingular points of $V(2n,2)$ with respect to a hyperbolic quadric. With the help of this bijection, we give a description of the strongly regular graph $N{O}_{2n}^{+}(2)$ in $V(2n,2)$. We also describe a graph with respect to a hyperbolic quadric in $V(2n,2)$ that was recently defined by Stanley and Takeda in $V(n,2)$. Similarly, we give a bijection between the point-hyperplane antiflags of $V(n,3)$ and the nonsingular points of one type in $V(2n,3)$ with respect to a hyperbolic quadric.