I recently learned about cross products and understood that cross products can be computed without an origin and coordinates in three dimensions, like vectors can be defined without coordinates.
But today I just found this statement in a calculus textbook: The cross product $A \times B$ is defined in three dimensions only. A and B lie on a plane through the origin.
Because of the remark about the origin above, I started to wonder if I got the basic concepts right.
This is my understanding. Based on the definition $\vec a \times\vec b=(||\vec a||||\vec b||\sin\Theta)\vec n$, we are able to compute cross products with information about magnitudes of two vectors, an angle between them and a direction given by the right-hand rule. Thus cross products can be defined without an origin and coordinates.
So I think the plane on which A and B lie above doesn't need to pass through the origin. Is it correct?
Your reasoning is correct. In fact, the essential part of a proper concept in geometry is the ability to define it without choosing particular coordinates.
The "3D-only" character of the cross product comes from the fact that in higher dimensions two vectors, and the plane which they share do not define exactly one perpendicular direction (like in the 3D case), as their are $N-2$ such directions where $N$ is the dimension. You can however define a product of $N-1$ vectors that will be a generalization of the cross product for a $N$-dimensional space. Then there is only one direcion left for the "N-cross product", so you're only left with the magnitude and orientation (pointing of the arrow) to define.
As for the magnitude of such a vector, the formula $\vec{a} \times \vec{b} = |a|\cdot |b| \sin{\theta}$ (I'm being very informal right now) comes from the fact, that we wish for the $|\vec{a} \times \vec{b}|$ to be equal to an area of a parallelogram with sides $a$ and $b$.
The orientation is chosen as such, that $\vec{a}$, $\vec{b}$ and $\vec{a} \times \vec{b}$ are like, consecutively: pointing finger, middle finger and thumb (AKA the right hand rule).
The proper mathematical definition is a bit messy and requires some linear algebra, and that's why usually textbooks struggle with how to present the subject while trying to be strict and often resort to define a "coordinate-only" version of the definition.
If you want to find out more about the formal algebraic definition, you can check for the wedge product: http://en.wikipedia.org/wiki/Wedge_product