The real linear space of matrices $$A=\begin{bmatrix}a&-c\\c&a\end{bmatrix},$$ with $a$ and $c$ real numbers, which satisfy the conditions (i) $\det(A)\geq0$ and (ii) $\det(A)=0\to A=0$, is a representation of the complex numbers. Is that true for all (special) spaces of $n\times n$ matrices satisfying these two conditions?



Are all special linear spaces representations of the complex numbers?
No: consider the 3-dimensional space of matrices $A=\begin{pmatrix}
0 & a & b & c\\ -a & 0 & c & -b\\ -b & -c & 0 & a\\ -c & b & -a & 0
\end{pmatrix}$. Then $\det(A)=\operatorname{Pf}(A)^2=(a^2+b^2+c^2)^2 $.


