late 12c., scathen, "to harm, injure, hurt; to cause harm, damage, or loss to," from Old Norse skaða "to hurt, harm, damage, injure," from Proto-Germanic *skathan- (source also of Old English sceaþian "to hurt, injure," Old Saxon skathon, Old Frisian skathia, Middle Dutch scaden, Dutch schaden, Old High German scadon, German schaden, Gothic scaþjan "to injure, damage").
In some sources this is traced to a PIE *sket- "to injure." The Germanic word was seen as cognate with some Celtic formations and Greek a-skēthēs "unharmed, unscathed," but Beekes finds that connection "impossible" on phonetic grounds and Boutkan, agreeing, writes that "The etymon is limited to Celt.-Gmc." and offers no IE etymology.
It survives mostly in its negative past participle unscathed, and in the figurative meaning "sear with invective or satire" (1852, usually as scathing). The latter seems to have developed specifically from the word in the sense of "scar, scorch" used by Milton in "Paradise Lost" (1667).