I've been searching online quite a bit for a simple and elegant geometric proof that the cross product distributes over addition. That is,
$$\mathbf a \times (\mathbf b + \mathbf c) = \mathbf a \times \mathbf b + \mathbf a \times \mathbf c.$$
Important: I do not seek a proof that derives from the formula for computing for the cross product based on vector components. I'm looking for a coordinate-free way of gaining intuition as to why the cross product distributes over addition this way using the intuitive "area of parallelogram and right-handed orientation" conceptual definition of the cross product.
Any thoughts, ideas, or insights?