Using dplyr and tidyr:
library(dplyr)
library(tidyr)
df <- as_data_frame(mat) %>% # convert the matrix to a data frame
gather(name, val, C_0:C_1) %>% # convert the data frame from wide to long
select(name, time, val) # reorder the columns
df
# A tibble: 6 x 3
name time val
<chr> <dbl> <dbl>
1 C_0 0.0 0.1
2 C_0 0.5 0.2
3 C_0 1.0 0.3
4 C_1 0.0 0.3
5 C_1 0.5 0.4
6 C_1 1.0 0.5