Is there a vector field $X$ on $\mathrm{M}_n(\mathbb{R})$ or $\mathrm{GL}(n,\mathbb{R})$ with the following condition: 
$$\begin{cases} X\cdot \mathrm{trace}=\mathrm{Det} \\X\cdot \mathrm{Det}=-trace \end{cases}$$
where $\mathrm{Det}$ is determinant?