Why do we need wedge product for $dxdy$ to polar coordinates when $dx$ and $dy$ are perpendicular to each other? Why can't we convert $dx$ and $dy$ to polar and straight out multiply the answer to get the area element? By multiply I mean, taking x=rcos(theta) and y=… and then taking differentials dx and dy and multiplying them. Answers here point out that wedge is supposed to be used but this is just a rectangle so do we still need it and why? Thank you.

As was pointed out in the comments to this question the multiplication of numbers and of differential forms are different. E.g. number multiplication obeys the commutativity rule $AB=BA$, while multiplication of differentials does not in general. That is why we use the $\wedge$-sign to indicate this different nature of multiplication.
Differentials have an anticommutativity rule: $dx\wedge dy=-dy\wedge dx$. Here is a conceptual way to understand this. Differentials like $dx$ with one component are used in measuring length, differentials like $dx\wedge dy$ with two components are relevant to measuring area. If you swap $dy\wedge dx$ then you change the "orientation" of the area in question and the area is measured with negative weight.
This comes in handy if you think about very elementary situations: If you have an L-shape that it is obtained by removing a rectangle from a larger one at one of the corners you want to compute the area of the L-shape as the difference of the areas of the two rectangles, i.e. the sum of the two areas but with the area of the smaller rectangle having negative weight. The anticommutativity of the $\wedge$-product helps in such situations and is useful there. Because of this, multiplication of differentials is different in nature than multiplication of numbers.
If you want to see how $\wedge$-products work out computationally in an explicit example you can go to the link provided by peek-a-boo or wikipedia.
You can also compare with multiplication of matrices or of quaternions: for these commutativity doesn't hold in general either. However the motivation and details why and how commutativity is violated in these cases is different from differentials.
