"staring around" is correct and usable in written English.
It is typically used to describe someone looking all around them without any particular focus. For example, "He was standing in the middle of the room, staring around nervously."
Across the road from the main square, a skinny man in a long leather jacket is staring around and barks instructions to another man.
On a night that was supposed to be all about Mitt Romney and his acceptance speech, the eighty-two-year-old actor and director delivered a rambling speech that had many of the attendees staring around in bewilderment.
I think this summarises the style because this book isn't written in a very formal way, and it doesn't focus on keeping checklists of birds or staring around and being obsessive about rare ones while ignoring all the more common species; instead it encourages you to look at all the birds, recognising their jizz, and getting to know them as well as you know your best friend.
"I just was staring around, dumbfounded in LA, like, What the fuck?" Sassone said.
I already did a bunch!" Agent Coke asks the group, staring around intently.
"Christ, are we in The Woman in the Fifth?" Paweł Pawlikowski asks, staring around.
Whenever she was around, staring her wide-eyed stare and talking aloud to nobody in particular, I kept stumbling and dropping things.
Don't stare at others, but don't wander around staring at your feet.
They walked around, staring up at all the jumbo television screens and at the only parental presence anywhere around: the huge, smiling faces of Charles Gibson and Diane Sawyer on a "Good Morning America" billboard.
Everyone stood around staring.
"It's not like people are turning around, staring at me on the street.
