"I've Got You Under My Skin" was written by Cole Porter in 1936 for the MGM musical Born to Dance.
It was nominated for the Academy Award for Best Song that year. Frank Sinatra first sang the song on his weekly radio show in 1946, as the second part of a medley with "Easy to Love". The song was largely forgotten by the time he "resurrected" it on his classic Songs for Swingin' Lovers album in 1956.
This version is the swingin' big-band version that builds to successive crescendoes due to the arrangement of Nelson Riddle. According to Sinatra: Behind the Legend, the "arrangement was so superb that after the musicians rehearsed it for the first time in the studio, they gave Riddle a standing ovation" (p. 190). Sinatra and Riddle rerecorded the song for Reprise in April 1963. Here is Sinatra performing it live on the 1965 television special Frank Sinatra: A Man and His Music (Nelson Riddle is conducting the orchestra, or "goosing butterflies" as Dean Martin described it):