Can the $\mathbb R$-algebra $M_n(\mathbb C)$ be generated using a set of only $2n$ of its elements?
My thoughts are that $M_n(\mathbb C)$ is a simple ring, and we are asking whether there is a surjective $\mathbb R$-algebra homomorphism $\phi:\mathbb R\langle X_1, \dotsc, X_{2n}\rangle \rightarrow M_n(\mathbb C)$. If this were to exist, then $M_n(\mathbb C)$ would be a simple representation of $\mathbb R\langle X_1, \dotsc, X_{2n}\rangle$. Unfortunately, the latter algebra is of wild type, and even its simple representations seem impossible to classify using any known method.