I have a project which I want to compile for Windows. I have the appropriate makefile, everything works with g++. I ran
apt-get install mingw32
If you see the ouput of
dpkg -L mingw32 you can see that the package contains a toolchain i.e. a set of alternative tools (compiler, linker etc) useful to compile your code in another system.
You have to use
/usr/bin/i586-mingw32msvc-g++ in order to compile your code. If I remember correctly exists a variable to tell the Makefile to use a toolchain like that.