This is also the space of real, symmetric bilinear forms in $\Bbb R^n$.

Two possible answers:
There are many other ways to characterize SPD matrices, but that's the only one I can think of at the moment that can be summarized as a single noun phrase. 


Note that this space is not a vector space, but is a convex cone in the vector space of nxn matrices (it is closed under addition and multiplication by positive scalars). Hence people sometimes refer to the "positive semidefinite cone". 


This is the symmetric space of GL_n(R) 


How about $M_n(\mathbb{R})^+$? I have seen $S^+$ or $S_+$ used to denote the set of positive linear transformations in a set $S$ of linear transformations on an inner product space, but this was in the context of operator algebras. 


For starters, since they're real I'd say symmetric instead of selfadjoint. 


It is often usefull to know that this set can be identified with the set of nonsingulat covariance matrices of random vectors with values in $\mathbb(R)^n$. 

