We were asked the following:
"Compute the dimension of the space of quadratic forms on $V=\mathbb{R^2}.$ Compute also the dimension of the space of symetric forms on $\mathbb{R^2}$, $S^2\mathbb{R^2}$. Compute also the dimension of the space of skew-symmetric forms $\Lambda^2\mathbb{R^2}$."
I'm not quite sure how to compute the dimension of these spaces. I wouldn't know how to do the only ways I know how, by using Rank-Nullity or finding a basis for the spaces.
To sum up the discussion in the comments: any quadratic form is of the form $ax^2+bxy+cy^2$, and so it can be uniquely written as a linear combination of $x^2$, $xy$, and $y^2$ (namely, with the coefficients $a$, $b$, and $c$). This means $\{x^2,xy,y^2\}$ is a basis, so the dimension is $3$.
For (skew-)symmetric forms you can do something similar. I don't know exactly what definition of them you are using, but you should be able to write them in a form where they manifestly depend on some number of parameters, and then by writing any form as a linear combination of some basic forms with these parameters as coefficients, you get a basis.