I've the following question $$ \text{Does it exist a basis of }M_n\left(\mathbb{R}\right)\text{ constituted of diagonalizable matrix ?} $$
My idea was to keep $E_{ii}$ on my basis, and maybe summing $E_{ij}+E_{ji}$ to have for sure a diagonalizable matrix but some vector of the basis are missing, how can I find them ?
The answer is yes and the field is unimportant. Note that when $i\ne j$, each $E_{ij}$ is the sum of two diagonalisable matrices: $$ \pmatrix{0&1\\ 0&0}=\pmatrix{0&1\\ 0&-1}+\pmatrix{0&0\\ 0&1}. $$ Therefore, the union of $\{E_{ii}\}$ and $\{E_{ij}-E_{\max(i,j),\,\max(i,j)}: i\ne j\}$ form a basis of the whole matrix space and each member in the basis is diagonalisable.