It was narrated from 'Amr bin Yahya that his father said to 'Abdullah bin Zaid bin 'Asim - who was one of the Companions of the Prophet (ﷺ) and the grandfather of 'Amr bin Yahya:
"Can you show me how the Messenger of Allah (ﷺ) used to perform Wudu'?" 'Abdullah bin Zaid said: "Yes. He called for (water for) Wudu' and poured some onto his right hand, washing his hands twice. Then he rinsed his mouth and nose three times, then washed his face three times, then he washed his hands twice, up to the elbows. Then he wiped his head with his hands, back and forth, starting at the front of his head and moving his hands to the nape of his neck, then bringing them back to the place he started. Then he washed his feet."
Arabic/English book reference: Vol. 1, Book 1, Hadith 98