From American colonial days to the early 1940s, soap was manufactured by an alkaline hydrolysis reaction called saponification.
Soap was made in huge kettles into which fats, oils, and caustic
soda were piped and heated to a brisk boil. After cooling for several days, salt was added, causing the mixture to separate into two layers with the "neat" soap on top and spent lye and water on the bottom.
The soap was pumped to a closed mixing tank called a crutcher where builders, perfumes, and other ingredients were added. Builders are alkaline compounds that improve the cleaning performance of the
soap. Finally, the soap was rolled into flakes, cast or milled into bars, or spray-dried into soap powder.
Source : http://www.epa.gov/ttn/chief/ap42/ch06/final/c06s08.pdf
Bleach is caustic, and will cause the saponification of fats and oils in your skin. Put simply, it's turning your skin into soap.
I echo Scooby's advice to wear gloves when handling the stuff.
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