Let $K$ be an antisymmetric matrix and $\varphi$ the application from $\left(\mathcal{M}_{2n,1}(\mathbb{R})\right)^{2}$ to $\mathbb{R}$ such that
$$\forall (X,Y) \in \left(\mathcal{M}_{2n,1}(\mathbb{R})\right)^{2}, \quad \varphi(X,Y) = X^{\top} K Y.$$
By computing $\varphi(X,X)^{\top}$ in two ways, show that $\varphi$ is alternating. Show similarly that $\varphi$ is antisymmetric.