If you change your time
column into row names, then you can use as.data.frame(as.table(mat))
for simple cases like this.
Example:
data <- c(0.1, 0.2, 0.3, 0.3, 0.4, 0.5)
dimnames <- list(time=c(0, 0.5, 1), name=c("C_0", "C_1"))
mat <- matrix(data, ncol=2, nrow=3, dimnames=dimnames)
as.data.frame(as.table(mat))
time name Freq
1 0 C_0 0.1
2 0.5 C_0 0.2
3 1 C_0 0.3
4 0 C_1 0.3
5 0.5 C_1 0.4
6 1 C_1 0.5
In this case time and name are both factors. You may want to convert time back to numeric, or it may not matter.