Derivative of bilinear forms

I want to solve the following problems:

1. Let $f:\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ be a bilinear form. Prove that it's differential is $$ Df_{(x,y)}(a,b) = f(x,b) + f(a,y).$$
2. Let $f:\mathbb{R}^3 \times \mathbb{R}^3 \to \mathbb{R}^3$ be the cross product funtion, that is, $f(x,y) = x \times y$. Calulate it's derivate in the point $(x,y)$.

I know the definition of differentiability for funtions $f:\mathbb{R}^n \to \mathbb{R}^m$. But here I am working with other funtions, so I don't know how to start.

Thanks.

For the first part,

\begin{align} Df_{(x,y)}(a,b) &= \frac{d}{dt}\bigg|_{t = 0} f((x,y) + t(a,b))\\ &= \frac{d}{dt}\bigg|_{t = 0} f(x + ta, y + tb)\\ &= \frac{d}{dt}\bigg|_{t = 0} (f(x,y) + tf(x,b) + tf(a,y) + t^2f(a,b))\\ &= f(x,b) + f(a,y). \end{align}

The bilinearity condition was used third line.

For the second part, show that $f$ is a bilinear form, and use the result in part 1 to find the derivative of $f$ at $(x,y)$.