It's date stamped, check the manufacture date page on the Remington Society of America for details.
1920
1918
1920 (no date code stamp)
1924, see related links.
January 6 1902-43,897 were made
Serial number 3273324 regards an early war (1941) built Remington Model 1903 Service Rifle, in caliber 30-06. Remington 1903A3 productions did not begin until 1942 dated to serial number 3348086 and higher.
I only have the serial list to 1931 and it's after that. It will however have a date code stamped on the barrel see related links for a page with details.
1920 (pre date code) and maybe even Model 12 on the receiver?
I checked www.oldguns.net and they show the manufacture date of Remington 1903A3 serial no. 3851292 as 1942.
RW isn't part of the serial no, it stands for Remington Works 1915
That is not a serial no of a Remington 1911A1 they stopped in 1945 with 2465139.
Remington Works - 1914 (late)