The modern senses were all around in Early Modern English, but Shakespeare's dominant use is "wild, excitable".
Boyet calls the ladies "mad wenches" (Love's Labour's Lost, II.i.243), and in The Two Noble Kinsmen Emilia describes men as "mad things" (II.i.180) and the Taborer addresses his fellows as "mad boys" (III.v.24).
The modern American sense is heard in Henry VI Part 3, when the Queen taunts York: "Thou shouldst be mad" (I.iv.89). And the modern sense of "crazy, weird" is used by Arcite when he asks Palamon "Is't not mad lodging,Here in the wild woods" (The Two Noble Kinsmen, III.iii.22).
The one usage which is no longer current is found in Othello, when Desdemona talks of her mother's maid: "he she loved proved mad" (IV.iii.26). Here the meaning is "wild", but with the important implication of "faithless, inconstant".
* David Crystal is author, with Ben Crystal, of Shakespeare's Words, published by Penguin. A subscription website is available at www.shakespeareswords.com