Does the space of $n \times n$, positive-definite, self-adjoint, real matrices have a better name? This is also the space of real, symmetric bilinear forms in $\Bbb R^n$.
 A: Two possible answers:


*

*Standard jargon is SPD (for "symmetric positive-definite"). 

*This isn't exactly a "name," but the n x n symmetric positive-definite matrices are exactly those matrices A such that the bilinear function (x, y) -> yTAx defines an inner product on Rn. Conversely, every bilinear function is of that form for some A, so with some abuse of terminology, you could equate the set of those matrices with the set of inner products on Rn. 
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. 
A: 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".
A: This is the symmetric space of GL_n(R)
A: 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.
A: For starters, since they're real I'd say symmetric instead of self-adjoint.
A: It is often usefull to  know that this set can be identified with the set of non-singulat covariance matrices of random vectors with values in $\mathbb(R)^n$.
