Was the last name Skakun exclusively Jewish?

Hello!
No, the surname Skakun is not necessarily Jewish, but rather Cossack.
Name Hannah is also not necessarily Jewish, the Ukrainian and Belarusian form of this name is Ganna (Ганна), which is transliterated into English as Hanna. The same name in Russian is Anna.

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And was it common for those of Jewish ethnicity in the Russian Empire to have their children baptized in the Orthodox churches

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- yes, it was quite common. Conversion to Orthodoxy provided an opportunity to avoid restrictions on education and career prospects for Jews. If you interested in this topic, I recommend to read about: Черта оседлости, Кантонисты, Выкресты
This article is about Jews in the Russian Empire https://arzamas.academy/materials/2203

Фамилия Скакун не еврейская. Большинство белорусы и украинцы. У меня родственники есть Скакуны - у них из Беларуси предки по этой линии, но имена все православные.
Ганна - это имя тоже у белорусов очень распространено, у меня в роду много по белорусской линии.

tIEWbTizk3xdUPYo написал:

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Ukrainian.

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The archive translated the birth certificate into Ukrainian. The name, which is written "Анна" in Russian, is written "Ганна" in Ukrainian. There is nothing Jewish in her first name, her last name, or the names of her parents.

ELM написал:

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The surname Skakun is not Jewish. Most are Belarusians and Ukrainians. I have relatives named Skakun - their ancestors on this line are from Belarus, but their names are all Orthodox.
Ganna is also a very common name among Belarusians; I have a lot of Belarusians in my family.

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Большое спасибо за ответ. Мой тест ДНК подтвердил, что я 100% восточный славянин, без еврейской генетики.