Skowron was born in Chicago, and nicknamed "Moose" not because he resembled the animal in any way, but because he had a short haircut as a child, which supposedly reminded his family of Italian dictator Benito Mussolini.
After playing college baseball at Purdue, Skowron signed with the Yankees in 1950 and became their more-or-less regular first baseman by 1955, although with manager Casey Stengel's complex platoon arrangements in those days, he never played more than 134 games in a season until 1960.
Skowron particularly shined in postseason play; in an era where "postseason" meant "World Series", he hit .293/.326/.519 in 39 games, with 39 hits and eight home runs. One of those home runs was particularly memorable; he homered and went 2-for-4 for the Dodgers in Game 2 of the 1963 World Series to help defeat his former team.
Later he played for his hometown White Sox and the Angels, finishing with an overall .282/.332/.459 line with 211 home runs. In recent years, Skowron worked as a community-relations representative for the White Sox, and occasionally appeared as a TV post-game analyst for the team.