When we had our son privately tested in order to advocate for early entry to first grade, the psychologist used the CTP (Comprehensive Test Program) from ERB http://www.erbtest.org/schools/achievement/ctp4
She used the end of first grade assessment. At the time he was still 3 months from starting school.

This fall, our school principal had the third grade teacher put together her own assessment of what she would expect a third grader to have mastered by the end of the year. The principal administered it and got to know our son and ultimately agreed that he needed a second acceleration into 3rd grade math.

You could go either with a standardized full scale assessment (preferred if you are looking for a full grade skip, I think) or just assess for a specific subject, if you are looking of subject acceleration, like we were this year.