The sky in the day is colored blue because molecules in the air scatter blue light from the light of the sun.
The sky during a some nights of having a full moon is very bright, as it shines the dark parts of the streets and cities on Earth. Also, this light, from the moon is reflected from the sun.
Since this light is still coming from the sun, why is thy sky in the night of a bright full moon not blue or even blueish?