Why is a bright night sky dark? 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? 
 A: The night sky under full moon does look dark blue (provided low light pollution).  However, the scattered "blue light" is not strong enough so that the night sky is still transparent to many stars.  Of course, it's difficult to see dim stars during full moon.  You may have better perception when you compare the background of full moon with thin clouds.
During day time, the blue sky may still transparent to half moon (of course between Moon rise and Moon set).  Moreover, you can see some bright stars just before Sun set or after Sun rise.  Think about the varieties of colours of sky during morning, noon and night from East, zenith and to West.
See the answers from another forum here and also Wikipedia article of Midnight blue.
A: Moonlight has a very similar spectrum to sunlight - it is slightly "pinker", but that isn't really relevant here.
However, the flux of light at the earth from the (full) moon is approximately 400,000 times less than from the Sun. This matters because the colour-sensitive receptor cells in your eye (known as cones) do not work well at low light levels.
If you actually  measure the spectrum  of the nighttime sky, produced by scattered moonlight,  it is in fact quite similar to that of the daytime sky, but the lower light levels mean the bluish colour is not perceived.
A: At night when the light is coming from sun, some part from it is transmitted, some is absorbed and rest is reflected by moon. So, the amount of light received by earth during night  of full moon(or any other night) is less than as received at daytime. In reality, during night we are not receiving sunlight.  The light we are receiving is the light which is reflected by moon; called moonlight. The phenomenon of scattering of light is exactly same during day and night time,  but at night time it occurs at low light levels, which doesn't results in blue colour of light. 
