Is there any lower bound known for the minimal number of generators needed to generate the full matrix algebra of real $nxn$ matrices - when using only symmetric matrices for the generators?

Analogous question for complex matrices - when using only Hermitian matrices for the generators.

I am aware that $3$ generators suffice when using only idempotent generators. This is a result of Naum Krupnik in 1992 (http://www.tandfonline.com/doi/abs/10.1080/00927879208824513)

I am not familiar with this type of results, so this might be well known or easy. Thanks for any tips.