Bases do not have to be ordered. I respectfully present the following as a counterpoint to Agustí Roig's response.
First of all, a basis is simply a set of vectors, by definition; and sets are unordered. If you wish to enumerate the vectors in a basis, you will of course do so in some order, but this order is arbitrary.
This arbitrary order of enumeration is the reason why we describe vectors as tuples. The order of the coefficients in a tuple matters only in as much as it must be in agreement with the arbitrary order which was selected for the basis vectors. To wit: let x, y be two arbitrary linearly independent vectors. The tuple [ 7 5 ] with respect to the (enumeration of the) basis v1 = x, v2 = y represents the same vector as the tuple [ 5 7 ] with respect to the (different enumeration of the same) basis v2 = x, v1 = y. The tuple is just a representation of a vector, relative to an arbitrarily chosen order for the basis.
Even more foundationally: "tuples" can be regarded as functions from the integers (e.g. the indices '1', '2', etc.) to the reals, complex numbers, or whichever set you draw your coefficients from. So [ 7 5 ] can be thought of as 'really being' the function mapping
'1' $\mapsto$ 7,
'2' $\mapsto$ 5.
Our enumeration of the basis has a similar role: choosing the enumeration v1 = x, v2 = y is equivalent to defining a mapping
'1' $\mapsto$ x,
'2' $\mapsto$ y.
Why do I have '1' and '2' in quotes? Because they're just labels; any other labels would do just as well. For instance, I could replace '1' and '2' with the vectors x and y themselves. Then we could define vectors by coefficient functions such as
x $\mapsto$ 7,
y $\mapsto$ 5;
that is, the coefficient 7 is associated to the vector x, and the coefficient 5 is associated to y, to represent the vector 7x + 5y. This is what we really mean anyway; never mind any ordering of the vectors in your basis. With this, the arbitrary ordering of the basis disappears, leaving nothing but what it is we really mean by it all.
So: when we order our bases, its only to make it easier to present things — it is not part of the definition, or part of the structure of the objects we really care about.
A morphism on a Euclidean space ought to preserve all of the structures of Euclidean geometry. In particular, it should map circles to other circles, to preserve the structures guaranteed by the third postulate; so it must be an isothety (a rigid transformation up to scaling). This includes, but is not restricted to, the rigid transformations.
Can you make any argument why the morphisms should consist only of rigid transformations (excluding isothetic maps such as v $\mapsto$ 2v)?