Alexander was born on the 13th of October, 1883, in St. Louis, Missouri, where he was also raised. He attended Washington University in St. Louis and graduated from there in 1905. While in college, Alexander was a member of the Missouri National Guard until 1908 and he then commissioned into the regular Army in 1916. When he deployed to Europe for the Great War, Alexander was a Captain and his actions during the Battle of the Argonne on the 26th of September, 1918, earned him the Medal of Honor. The citation reads:

Unwilling to sacrifice his men when his company was held up by terrific machine-gun fire from iron pill boxes in the Hindenburg Line, Capt. Skinker personally led an automatic rifleman and a carrier in an attack on the machine guns. The carrier was killed instantly, but Capt. Skinker seized the ammunition and continued through an opening in the barbed wire, feeding the automatic rifle until he too, was killed.

Alexander Rives Skinker was 34 years old when he died and the War Department issued his posthumous Medal of Honor on the 18th of January, 1919. He is buried in the Bellefontaine Cemetery in St Louis, Missouri: Block 78/79, Lot 2342.