  In this paper, we present a new formula of the determinant tensor $det_n$ for
$n \times n$ matrices. In \cite{kim2023newdet4}, Kim, Ju, and Kim found a new
formula of $4 \times 4$ determinant tensor $det_4$ which is available when the
base field is not of characteristic $2$. Considering some symmetries in that
formula, we found a new formula so that \begin{equation*}
  \operatorname{Crank}(det_n) \leq \operatorname{rank}(det_n) \leq
\frac{n!}{2^{\lfloor(n-2)/2 \rfloor}} \end{equation*} when the base field is
not of characteristic $2$.
