Open-air electron tube
The electron wouldn't get very far, the mean free path in air would be very short or around nm's before it would collide with an air particle. This would lower the current of the 'open air' electron tube significantly. You might be able to overcome this by shortening the distance between anode and cathode or by raising the voltage of the 'tube'. 

The point of a vaccum tube is to enhance the mean free path of the free electrons, and you also need a heat source to 'boil off' electrons. Air would cool down the metal and lower the mean free path. 

A gas filled tube works with a similar principle, but uses ions instead of electrons to create a current and they operate in the breakdown region of the gas.

If you wanted to create a breakdown device that is ionizing the are here is some more info:


Air has a very high impedance so the current would be in the pA to fA range, so you would end up having to amplify the current anyway which would kind of defeat the purpose.
The humidity in the air would also vary greatly, and would cause the 'open air' tube to be dependent on humidity and temperature.