بِسْمِ اللَّهِ
In the name of Allah.
The prefixed bāʾ is a preposition, and ism is governed by it. The expression is understood with an implicit verb. A common reconstruction is أَبْتَدِئُ بِاسْمِ اللَّهِ: “I begin in Allah's name.” In recitation, أَقْرَأُ بِاسْمِ اللَّهِ, “I recite in Allah's name,” suits the action.
Leaving the governing expression unstated makes the opening suitable for many acts. The relevant act is supplied by the context; one fixed reconstruction is not required in every situation. Giving precedence to Allah's name also makes the invocation prominent.
Example: At the start of writing the implicit act is writing, while at the start of recitation it is reciting.