Minors or Cofactors, why not in this way?

Let us consider $M$ one $n\times n$ symmetric matrix. Its is very well know that

\begin{equation}\label{detM}\det M=\sum_{i=1}^n(-1)^{i+J}m_{Ji}M_{Ji},\end{equation}

where $M_{Ji}$ is a so-called minor or cofactor of $M$, obtained by taking the determinant of $M$ with row  $i$and column $J$ "crossed out."

Also the determinant of the matriix $M$ can be write in the form

\begin{equation}\label{detM1}\det M=m_{1i_1}\cdots m_{ni_n}\varepsilon^{i_1\cdots i_n}\end{equation}

that  can be rewrite as

\begin{equation}\det M=\frac{1}{n!}\varepsilon^{a_1\cdots  a_n}m_{a_1i_1}\cdots  m_{a_ni_n}\varepsilon^{i_1\cdots i_n}\end{equation}

or as

\begin{equation}\label{detM2}\det M=m_{11}\gamma^{11}\cdots  m_{1n}\gamma^{1n},\end{equation} where we define

\begin{equation}\label{cof}\boxed{\gamma^{IJ}\equiv\frac{1}{(n-1)!}\varepsilon^{Ii_2\cdots i_n}m_{i_2j_2}\cdots m_{i_n j_n}\varepsilon^{Jj_2\cdots j_n}}\end{equation}

with the nice definition for the Minor or Cofactor.