I've found a Henry Skinner born Q3 1871 in Bethnal Green that's a possibility, no middle name given:
Skinner, Henry Bethnal Green 1c 279 Q3 1871
Having checked 12 quarters in all from Q1 1871 to Q4 1873 by looking at the original registers on The Genealogist, he's the only Henry Skinner born in Bethnal Green so if you're sure he was born around that time then it's a pretty good bet I'd say.
I also found a Henry Skinner in the 1881 census aged 10, born London Middlesex living in Clerkenwell (close to Bethnal Green) with his mother Fanny (a widow), elder brother William aged 13 and younger siblings. I looked for a marriage for Daniel Skinner between 1850 and 1899 to catch all possibilities and there is one in Bethnal Green in Q3 1874 with possible brides named Amelia, Louisa or Elizabeth. There is one who married a Fanny but as its in Q4 1851 it's too early to be the Fanny in the census above.
I think your best bet would be to purchase the birth certificate mentioned earlier.
Heather