'allow me to explore' is correct and usable in written English.
You can use it when you are asking permission to explore something (e.g. a place, an idea, a concept, etc.). For example: "Allow me to explore the possibilities of creating an innovative way to solve this problem."
I sort of steered his life towards places or encounters or events that would allow me to explore my prejudices and my likes.
On weekends, I would take short sightseeing trips, or participate in some activities that allow me to explore more of the Japanese culture.
But the society sees them as different, so I was looking for a story that would allow me to explore this topic in its whole complexity.
To make clearer the value of online discussion, allow me to explore two different modes of schooling: the online school and the traditional brick-and-mortar school.
"I have many formal concerns which allow me to explore my personal ideas of self, family history, and the identities we create online".
These are chances to experiment and understand the technology, whereas my personal projects allow me to explore more conceptually and make uncompromised work that points towards the kinds of stories and ideas that really define my voice.
There's the rub, for me at least: I signed on with an untenured professor who, although he probably wouldn't say this, has no interest in allowing me to explore my interests unless those explorations will directly--and definitely--benefit him.
The EP allows me to explore that side.
" 'Carnation' allowed me to explore minutiae onstage," Ms. Coates said.
Photography allows me to explore places without having a set agenda.
Although my programme is entirely invented, writing about it allowed me to explore how I felt about certain journalistic practices.
