number of pairwise present data in matrix with missings

A <- matrix(1, 6, 3)
A[1,1] <- A[3, 1] <- A[2,2] <- NA
A

crossprod(!is.na(A))

is there a smart way of determining the number of pairwise present data
in a data matrix with missings (maybe as a by-product of some
statistical function?)

so far, i used several loops like:

for (column1 in 1:99) {
 for (column2 in 2:100) {
   for (row in 1:500) {
     if (!is.na(matrix[row,column1]) & !is.na(matrix[row,column2])) {
       pairs[col1,col2] <- pairs[col1,col2]+1
     }
   }
 }
}

but this seems neither the most elegant nor an utterly fast solution.

thanks for suggestions.
andreas wolf