How Can We Achieve the Tensor Product of Two Vector Spaces?

[ShayanJ]

I'm trying to understand tensor product of vector spaces and how it is done,but looks like nothing that I read,helps!
I need to know how can we achieve the tensor product of two vector spaces without getting specific by e.g. assuming finite dimensions or any specific underlying field.
Another thing,is it possible to define it with no reference to any basis of the vector spaces?
Thanks

 

[Fredrik]

This thread should be useful. See in particular posts 1,2,8,9. Note that when quasar987 corrects a mistake I made in post 1, he's referring to a mistake that was later fixed in an edit. So post 1 should be OK.