When someone lays eyes on you and keeps looking, it is a gaze or a stare. You can gaze back or just say, "Take a picture; it lasts longer."
Gaze is both a verb and a noun. When people daydream, they often gaze off into the distance without focusing on anything. Another type of gaze, though, is to look very intently at something, as when you gaze at the sky making pictures out of clouds. A contest to see who will blink first means taking a long gaze into someone's eyes, and this type of gaze (the noun) will leave your eyes in a bit of a haze when you're done.