Summarizing elements for a data.frame

Hello again, let say we have following data:

Dat1 <- structure(list(factor.sample.LETTERS.1.3...6..replace...T.. =
structure(c(1L,
3L, 2L, 1L, 3L, 3L), .Label = c("A", "B", "C"), class = "factor"),
    factor.sample.letters.1.2...6..replace...T.. = structure(c(2L,
    2L, 1L, 1L, 2L, 1L), .Label = c("a", "b"), class = "factor"),
    X1.6 = 1:6), .Names = c("factor.sample.LETTERS.1.3...6..replace...T..",
"factor.sample.letters.1.2...6..replace...T..", "X1.6"), row.names = c(NA,
-6L), class = "data.frame")

Out of this data.frame, I want to create a Table with rows coming from
1st column of Dat1 and columns are coming from 2nd column of Dat1 and
each entry will be the sum for 3rd column of Dat1, i.e. the element
for (1,1) will be sum for all element in 3rd column corresponding to
(A, a) and so on.

I tried with table() however could not achieve what I wanted.

Can somebody give me some pointer?

Thanks and regards,