If you are using a cleaning agent (called White King or Demostos as used in Australia) containing a chlorine base (chemical name Sodium Hypochlorite), use white vinegar on your hands and the smell will disappear.
By Spoilground from Portarlington, Australia
Editor's Note: Sodium Hypocholorite is also commonly known as bleach.
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How do I get rid of chlorine bleach smell from my basement? It is coming up into my house and we can't breath, I used too much to get rid of mildew!