Name/notation for group of permutation matrices $S_n$ is the group of permutations, and there is a bijection between each permutation and its permutation matrix. Is there a name/notation for the group of permutation matrices?
 A: It’s simply called : the group of permutation matrices. 
A permutation matrix associated with the bijection $\tau$ is often denoted : $P_\tau$.
A: I cannot say that I have ever heard the group of permutation matrices called anything other than $S_{n}$ but most define $S_{n}$ as the set of bijections on an $n$-element set and there is obviously quite a few $n$-element sets out there. The common choice is thinking of bijections on the set $\{ 1 ,\dots, n\}$. I think this is probably best for any type of calculation to be carried out by hand especially considering you can write elements rather compactly using cycle notation, but you could of course use a different $n$-element set. 
We could use other symbols as well obviously, such as the elements of the canonical basis for $F^n$ for some field $F$ but it obviously depends on context. Thinking about $S_{n}$ as the set of bijections on the canonical basis for $F^n$ obviously has broader implications because there is additional structure there. However, could you imagine working out orbits, stabilizers, cosets, etc. by hand if you were working exclusively with matrices? That would get really cumbersome for $n > 2$. If you are using a computer algebra system though, then this is of course turns out to be a great way to do things. 
