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Obsess someone or affect someone's deep feelings, as in Jean's really gotten under his skin; he misses her terribly . Cole Porter used this sense in his love song, “I've Got You Under My Skin” (1936).

Irritate someone, as in She really knows how to get under my skin with her nagging . This expression no doubt alludes to burrowing or stinging insects that cause itching or similar skin irritations. [Late 1800s]