I was trying to prove that $$GL(m,\Bbb{C}) \subset GL\_+(2m,\Bbb{R})$$

If we treat the complex matrix $W\in GL(m,\Bbb{C})$ as $$W=
\begin{bmatrix}
W\_r & -W\_i \\ W\_i &W\_r
\end{bmatrix}\in R ^{2m\times 2m}
$$

I have no idea how to prove this I try to use the inverse formula for the block matrix however $W\_r$ needs not to be invertible therefore the formula below does not work:

$$ \det {\begin{pmatrix}A&B\\C&D\end{pmatrix}}=\det(A)\det \left(D-CA^{-1}B\right).$$

I try to do some fundamental transformation to convert the $2m \times 2m$ matrix into triangle matrix, but it seems not very clear how to find such transformation.I try to analysis the eigenvalue of the $2m\times 2m$ matrix is seems not very clear.There must be many different ways to prove this result, but no idea in my mind.