How do I export map of large area in .osm format?

The Export tab on the main OSM site is not the appropriate way to export large areas: it uses too many resources from the API. I also think it is limited by constraints in the API interface itself: there is a maximum area and maximum amount of data which can be downloaded through the API map call.

Therefore you will need to use the available alternatives:

• One of the XAPI instances. These are designed for providing read-only access to very recent snapshots of OSM data. Unfortunately, they are often overwhelmed by requests. The JAXPI instance run by MapQuest is probably the best one to use: it should certainly cope with downloading a city.
• Using an already available download, such as those of Geofabrik and Cloudmade. These cover many (but not all) countries and most regions of the world. For places which are very actively mapped there may be extracts of smaller areas. In most cases you will still want to reduce the amount of data you use. To do this you should use Osmosis to read the extract file and extract a bounding box from it (osmosis --rx file="bla.osm" --bbox left=##.## right=##.## top=##.## bottom=##.## --wx file="extracted_city.osm", where the numeric values are the lat and lon of a bounding box)