This is an extremely basic and short tutorial. A skybox is needed in every map. Without it, your map will "leak" and will thus cause errors while compiling. This means that you will not be able to play the map.
Select the caulk texture. Draw a big block over your map. Everything you want in the map should be in the inside of this blok. With the block selected, use the hollow option. The brush will be hollowed. Each of the six sides will be a single brush.
Select all of the brushes, one at a time, and make sure they don't overlap eachother.
Go to the texture menu and select the sky/skies map. You will now be able to pick the sky of your choice.
Press ctrl and shift at the same time and select all of insides (the sides and the upper sky). Paint them by selecting the sky texture of your choice while the sides are selected.
Note: Do note that by using the ctrl and shift option you will only paint the selected side of the brush, which is what you want. Also note that the brush under your map does not neet to be painted with a sky texture.
You now have a sky, but you still need the light in the map. With light I mean sunshine etc. To do this, consult the worldspawn tutorial.