"a look about" is correct and can be used in written English.
You can use it to mean looking around for something. For example, "Sarah took a look about the room, trying to find her keys."
"They sniff around, they might sit down, have a look about, and then they'll eventually climb up the tree and have a nibble.
Locals show up with a horse, have a look about, possibly swap their horses for someone else's, then go home.
There's a look about him – not vacant, but distant.
She must have had a look about her, because his expression grew more guarded.
The tiny space had a look about it of "Pee-wee's Playhouse," or something imagined by Rodney Alan Greenblat, another '80s-era artist.
But first a meal in the Museum of Wine beneath the casino, and a look about their spectacular cellar and that bottle of wine, an 1899 vintage worth £16,000.
Burroughs even thought that his friend had "a look about him... of the gray, eternal sea that he so loved, near which he was born, and that had surely set its seal upon him".
Take a look after about an hour.
Tall, white, with receding grey hair, brown eyes and a prominent nose, he looked about 70.
Well, that has a look of truth about it.
There's also a look of Ikea about some of the furniture.
