Show for any k that $\|x\|^2\|y\|^2 -( \textbf{x} \cdot \textbf{y})^2 = \frac{1}{2} \sum_{i, j=1}^k (x_i y_j - x_j y_i)^2$

Question

I am doing some practice tasks on Cauchy-Schwarz inequality before my university classes start and I am faced with this problem. I simply have no idea how I would go about showing this, I am completely blank and I would appreciate it if someone could show me how to prove this.

$\|x\|^2\|y\|^2 -( \mathbf{x} \cdot \mathbf{y})^2 = \frac{1}{2} \sum_{i, j=1}^k (x_i y_j - x_j y_i)^2$

Answer 1

Note that $$\sum_{ij}(x_iy_j-x_jy_i)^2+2(x\cdot y)^2=\sum_{ij}((x_iy_j-x_jy_i)^2+2x_ix_jy_iy_j)=\sum_{ij}((x_iy_j)^2+(x_jy_i)^2)=2x^2y^2.$$