  In this paper first we describe all (not necessarily linear or bijective)
transformations on $\mathbb{R}^d$ with $2\leq d<\infty$ which preserve the area
of parallelograms spanned by any two vectors. We also characterize those (not
necessarily linear) bijections on an arbitrary real Hilbert space that preserve
the latter quantity. This answers a question raised by Rassias and Wagner, and
it can be considered as a variant of the famous Wigner theorem on real Hilbert
spaces which plays an important role in quantum mechanics. As a consequence, we
solve a preserver problem of Moln\'ar and Timmermann which has remained open
stubbornly only in the two-dimensional case. Finally this two-dimensional
result will be applied in order to strengthen their theorem in higher
dimensions.
