‘Look me in the eye’ is a frequently used idiomatic expression in British English (and American English too).
‘To talk to someone in an honest way that shows no doubts’ (http://dictionary.cambridge.org).
‘Look directly at someone without showing embarrassment, fear, or shame’ (https://en.oxforddictionaries.com).
It would seem to make more sense to say ‘look me in the eyes’ because most people have two eyes but it’s an idiomatic expression and idioms don’t make sense.
As an alternative to using the idiom, a mother might say to her young son: ‘Look into my eyes Jimmy and tell me honestly; did you kick the cat’?
Using the idiom form she might say: ‘Look me in the eye Jimmy and tell me honestly; did you kick the cat’?
We also say things like ‘keep your eye (yes that’s one eye again) on my handbag while I get another another drink (so that nobody steals it)’.
If you are playing a sport such as tennis and you keep missing the ball with your racquet, your coach might say ‘keep your eye (one eye again) on the ball (so you don’t keep missing it)’.
In the handbag example it’s also ok to substitute the word ‘an’ for ‘your’. So you can say ‘keep an eye on my handbag’.