Prove using Jacobi's second theorem on determinants that

$$\begin{vmatrix} 0 & a & b & c \\ -a & 0 & d & e \\ -b & -d & 0 & f \\ -c & -e & -f & 0 \\ \end{vmatrix} = (af-be+cd)^2$$

I can easily prove it using Laplace expansion for determinants but have no idea how to prove it using Jacobi's second theorem. A corollary of the theorem tells that the determinant would be a perfect square of a polynomial of the elements. But nothing further. Any help will be appreciated.

For the theorem, have a look here: Jacobi's Second Theorem on Determinants


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