How to create e dates in R from year, month, day?

Question

I have a dataset where the creator saved the date in three seperate variables. My goal is to convert these three date variables into a single numerical variable that keeps the chronological order. To do that, I first pasted together my 3 variables. However, I`m seemingly unable to convert this into a numeric variable. See below for some example code and the output I want. There must be some easy way to do this in R but everything I try results in errors.

#example data
day <- c(1,2,13)
month <- c(1,2,12)
year <- c(2020, 2020, 2020)


#paste variables
Time <- paste(year, formatC(month, width = 2, flag = 0), formatC(day, width = 2, flag = 0))

WantedOutput <- c(20200101, 20200202, 20201213)