Secondary Navigation inside sidebar widget

I would like to create a dropdown select box inside a sidebar widget.

As it appears difficult to create a select box that looks the same across all browsers, I wondered if it possible to make the secondary navigation appear as a select box in the sidebar.

I know that the secondary navigation can be positioned in the sidebar but I would like it inside a widget as it will have text above it. Also it is possible to change the sub-navigation items to drop below instead of to the side?

Unless you know of a fairly easy way to create a select box that would look the same across all browsers?

Many thanks.