Throughout the rest of the problem, $K = J_{n}$. For all $X = \left(\begin{array}{c} x_{1} \\ x_{2} \\ \vdots \\ x_{2n} \end{array}\right) \in \mathcal{M}_{2n,1}(\mathbb{R})$ and for all $Y = \left(\begin{array}{c} y_{1} \\ y_{2} \\ \vdots \\ y_{2n} \end{array}\right) \in \mathcal{M}_{2n,1}(\mathbb{R})$, where $\varphi(X,Y) = X^{\top} J_{n} Y$, show the equality
$$\varphi(X,Y) = \sum_{k=1}^{n} \left(x_{k} y_{k+n} - x_{k+n} y_{k}\right).$$