THE IOWA of Test of BAsic Skills is a test you can administer at home. One source is Brewer Testing Service. Others use BJU Testing Service. There are several out there. You get it in the mail, administer the test, mail it in for scoring, and receive a score report.
I think for actual grade level placement, schools would probably like to administer their own end of year exams in the various subjects w/ the student required to meet some score - 80% correct or 90% correct etc.
Or schools might just place by age. If you wanted to skip, that would be a horse of a different color.