The only way to know you know nothing is to know something then admit what you know is susceptible to constant change then conclude you know nothing. Socrates was suggesting that wisdom begins by acknowledging one's limitations; understanding that there must exist a vast ocean of undiscovered truth. I don’t think he literally meant to suggest that we don’t know anything at all.
Socrates wasn't supporting the idea of recognizing one's ignorance as wise. He was demonstrating that ignorance is the beginning of knowledge. In order to acquire knowledge, you must first realize where the gaps in your knowledge lie. Then fill them up. A person who continues to know nothing when knowledge is available is an ignoramus.