I am looking for, a couple of days, to download the elements through the asset store, but when I go to open (either via browser or via Unity) the asset store hangs in this gray screen:
And then I can’t download anything.
Do you have any suggestions on how to fix it?
Make sure you don’t run Unity in DX11 mode. It’s a known bug that the Asset Store won’t load properly with DX11 enabled. Here’s a Guide on how to enable/disable the mode: http://docs.unity3d.com/Manual/DirectX11.html
Edit: I just read here that this should be fixed with Unity 4.5.3.
you’re right, it was the fault of DX11!
thanks so much 
According to the bug tracker, this should be fixed with Unity 4.5.3!
May I ask which version you are running? (see Help > About Unity… inside Unity)
version 4.5.5.f1