Faster matrix multiplication by replacing a double loop

I have a dataframe which looks a bit as produced by the following code (but much larger)

set.seed(10)    
mat <- matrix(rbinom(200, size=1, prob = .5), ncol = 10)

In the columns are issues and 1 indicates that an observation is interested in a specific issue. I want to generate a network comparing all observations and have a count of issues that each dyad is jointly interested in.

I have produced the following code, which seems to be working fine:

mat2 <- matrix(NA,20,20)

for(i in 1:nrow(mat)){
    for(j in 1:nrow(mat)){
       mat2[i,j] <- sum(as.numeric(mat[i,]==1) + as.numeric(mat[j,]==1) == 2)
    }
 }

So I compare every entry with every other entry, and only if both have a 1 entry (i.e., they are interested), then this sums to 2 and will be counted as joint interest in a topic.

My problem is that my dataset is very large, and the loop now runs for hours already.

Does anyone have an idea how to do this while avoiding the loop?