Can you say I didn't want to get stuck in someone's eyes when you don't want to be seen by someone? What is the other expression of avoiding seeing by someone or stay invisible?
That sounds very unnatural and not idiomatic. Possible alternative idioms:
keep under the radar (very informal, won't find it in most dictionaries):
doing something without other people noticing.
A: Wow! What does her father think?
B: Well, we’re keeping our relationship under the radar because we don’t want him to know.
lay low (very informal, colloquial, and technically incorrect grammatically):
- To keep oneself or one's plans hidden.
American Heritage Dictionary
"Lie low" would be grammatically correct, buy "lay low" to stay out of sight is more commonly heard in my opinion.
steer clear of TFD
Stay away from, avoid
I was hoping to steer clear of his family at our friend's funeral.