Sum of all elements in a matrix

I don't know if it has a nice name or notation, but for the matrix $\mathbf A$ you could consider the quadratic form $\mathbf e^\top\mathbf A\mathbf e$, where $\mathbf e$ is the column vector whose entries are all $1$'s.

Using the sum of all elements does not contain any information about endomorphisms, which is the reason why you will not find such an operation in the literature.

If this is interesting enough, you can get the sum of all squares using the scalar product $$ \phi(A,B) := \mathrm{tr}(A^T B)$$ In fact $\mathrm{tr}(A^T A) = \sum_{i=1}^n \sum_{j=1}^n a_{i,j}^2$

The term "grand sum" is commonly used, if only informally, to represent the sum of all elements.

By the way, the grand sum is a very important quantity in the contexts of Markovian transition matrices and other probabilistic applications of linear algebra.

Regards, Scott