s there a R function that applies a function to each pair of columns?

It wouldn't be faster, but you can use outer to simplify the code. It does require a vectorized function, so here I've used Vectorize to make a vectorized version of the function to get the correlation between two columns.

df <- data.frame(x=rnorm(100),y=rnorm(100),z=rnorm(100))
n <- ncol(df)

corpij <-function(i,j,data){cor.test(data[,i],data[,j])$p.value}
corp <- Vectorize(corpij, vectorize.args=list("i","j"))
outer(1:n,1:n,corp,data=df)