Probably a well-know question, but I haven't solved it, so I'll ask.

I can show that every matrix in $M_2(\mathbb{R})$ is the sum of two squares of matrices in $M_2(\mathbb{R})$. If $n>2$, I can also show that every matrix in $M_n(\mathbb{R})$ is the sum of three squares of matrices in $M_n(\mathbb{R})$.

So my question is : Is every matrix in $M_n(\mathbb{R})$ is the sum of two squares of matrices in $M_n(\mathbb{R})$ (n>2)?