Is there a way to keep my map visible while I walk around?

One of the features I really liked in Diablo 2 was the ability to change your map position. You could either keep it in the upper right hand corner or switch it so that it was over-layed on the screen and your character could essentially walk around on top of it. Like this:

Is there a way that I can set my map up like this in Diablo 3? If not exactly like this, is there any option other than keeping it in the upper right hand corner?

No, that feature was intentionally taken out by the designers because they wanted to avoid that situation we all know in D2 where we spend about 10% of our time looking at the action and 90% looking at the minimap.

Source:

You are very much correct in your statement that this was a deliberate design choice, Caelestis. We did not want to have the Diablo II scenario where people just stared at a dot on a map to guide themselves through an area. We made the map full screen, completely opaque and prevented commands from being sent while it is open to stop this from occurring. We also like the concept that your character has to take a few moments and stop so that they can get their bearings :)

As a bonus, you can always kind of see what areas you haven't explored yet in the minimap to the upper right of the screen.