No it is not normal to have a red cross when connected by wireless. The little monitor icon should turn into a signal strength bar graph when wireless is connected instead of the wired icon of a small monitor.
Sometimes the taskbar icon does not truly represent the actual connection situation.
I suggest you right click on the taskbar icon and select "Open Network and Sharing Centre". Then select "Change Adaptor Settings". Click on the connection being used and select "Disable". Leave it a few seconds the click on Enable. It should then re-search for an IP and probably will then show the correct status, at least it works for me when I occasionally have the taskbar icon showing the spinning circle although I have internet access.