How to derive the covariance matrix from a rotated ellipse?

Question

I'd like to derive the covariance matrix that defines a given ellipse.

Information I have:

1. length of major axis $\lambda_1$
2. length of minor axis $\lambda_2$
3. angle of rotation of the ellipse is $\theta$ degrees
4. determinant of the desired covariance matrix is $ = 1$

Given the four sets of the information above, is there a way to go backward and derive the covariance matrix that defines the ellipse?

Thank you in advance!