Old English gat "she-goat," from Proto-Germanic *gaito (source also of Old Saxon get, Old Norse geit, Danish gjed, Middle Dutch gheet, Dutch geit, Old High German geiz, German Geiß, Gothic gaits "goat"), from PIE *ghaid-o- "young goat," also forming words for "to play" (source also of Latin hædus "kid").
They are sprightly, capricious, and wanton, and their strong odor (technically called hircine) is proverbial. [Century Dictionary]
The word for "male goat" in Old English was bucca or gatbucca (see buck (n.)) until late 1300s shift to he-goat, she-goat (Nanny goat is 18c., billy goat 19c.). Meaning "licentious man" is attested from 1670s (hence goat-milker, name of a bird formerly believed to suck the milk from goats at night, but also old slang for "a prostitute," also "the female pudendum"). To get (someone's) goat is by 1908, American English, the source of many fanciful explanation stories; perhaps from French prendre sa chèvre "take one's source of milk," or more likely it is "to steal a goat mascot" from a racehorse, warship, fire company, military unit, etc.
... to become separated from your goat is a thing no soldierman is willing to contemplate. ["Letitia, Nursery Corps, U.S.A.," in American Magazine, vol. lxiv, June 1907]
skin n.
c. 1200, "animal hide" (usually dressed and tanned), from Old Norse skinn "animal hide, fur," from Proto-Germanic *skinth- (source also of rare Old English scinn, Old High German scinten, German schinden "to flay, skin;" German dialectal schind "skin of a fruit," Flemish schinde "bark"), from PIE *sken- "to peel off, flay" (source also of Breton scant "scale of a fish," Irish scainim "I tear, I burst"), extended form of root *sek- "to cut."
The usual Anglo-Saxon word is hide (n.1). The meaning "epidermis of a living animal or person" is attested from early 14c.; extended to fruits, vegetables, etc. late 14c. Jazz slang sense of "drum" is from 1927. As short for skinhead from 1970. As an adjective, it formerly had a slang sense of "cheating" (1868, compare the verb); that of "pornographic" is attested from 1968. Skin deep "superficial, not deeper than the thickness of the skin" (also literally, of wounds, etc.) is attested by 1610s:
All the carnall beauty of my wife, Is but skin-deep.
[Sir Thomas Overbury, "A Wife," 1613; the poem was a main motive for his murder]
The skin of one's teeth as the narrowest of margins is attested from 1550s in the Geneva Bible, a literal translation of the Hebrew text in Job xix.20. To get under (someone's) skin "annoy" is from 1896. Skin graft is from 1871. Skin merchant "recruiting officer" is from 1792 (the older sense is "dealer in hides"). Skin and bone as a description of emaciation or extreme leanness is in Middle English:
Ful of fleissche Y was to fele, Now ... Me is lefte But skyn & boon. [hymn, c. 1430]