It is a standard fact that everyt real $n$-dimensional algebra is a subalgebra of $M_n(\mathbb R)$.

The transposition map, operating in $M_n(\mathbb R)$, is an involutive ($(A^t)^t=A$) antiautomorphism (an $\mathbb R$-linear isomorphism satisfying $(AB)^t=B^tA^t$). This makes $M_n(\mathbb R)$ a real *-algebra. The same is true in every transpose-closed subalgebra of $M_n(\mathbb R)$.

Is every real *-algebra of this kind? (a transpose-closed subalgebra of $M_n(\mathbb R)$, with the * represented by transposition)

This is true at least in the famous real *-algebras $\mathbb C$ and $\mathbb H$.