Is it true to say that every matrix $A\in M\_n(\mathbb{R})$ is similar (conjugate) to a matrix $B=(b\_{ij})$ with $b\_{ij}=-b\_{ji}$ for all $i\neq j$?(With some abuse of terminology,a matrix $B$ with this property is called "Semi antisymmetric"). 



Is every real matrix conjugate to a semi antisymmetric matrix?
Yes. Every matrix can be written as the sum of a symmetric plus an antisymmetric one: $A = \frac{A+A^T}{2}+\frac{A-A^T}{2}$. Now change basis such that the symmetric part is diagonal.


