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.

Definitions of gaze
  1. noun
    a long fixed look
    “he fixed his paternal gaze on me”
    synonyms: regard
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    type of:
    a fixed look with eyes open wide
  2. verb
    look at with fixed eyes
    synonyms: stare
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    outface, outstare, stare down
    overcome or cause to waver or submit by (or as if by) staring
    observe the stars
    type of:
    perceive with attention; direct one's gaze towards
Word Family