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How do you find another ordered basis for a vector space $C^n$ over the field $R$ if one of the basis is given? That is, I know the standard basis for this vector space over the field is:

$\{ (1,0,...,0),(i,0,...0),.....,(0,0,...1),(0,0,....i) \}$.

But, I want to find another basis for this vector space and this new basis should not contain an $R$-scalar multiple of a vector in the standard basis. I need this information in order to determine an invertible matrix $P$ such that for any vector $v$ in $C^n$, $[v]$standard basis $=$ $P [v]$new basis.
Thanks for the help!

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    I'm confused. You refer to both C^n and R. Do you mean $n$-dimensional complex space $\mathbb{C}^n$ and the real numbers $\mathbb{R}$? Are you thinking of $\mathbb{C}^n$ as a vector space over $\mathbb{R}$ or over $\mathbb{C}$? – 2017-02-16
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    Its the n-dimensional complex space over the real numbers. – 2017-02-16
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    In that case, keep in mind that $\mathbb{C^n}$ is actually $2n$ dimensional as a vector space over $\mathbb{R}$, not $n$ dimensional. – 2017-02-16

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Say we have a field $F$. Then any $n$-dimensional vector space over $F$ is isomorphic to $F^n$. If you have a nonsingular square matrix with entries in $F$, then its rows are linearly independent and hence form a basis for $F^n$. That's a way to get new bases.

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    Can you provide an example on how to do that? – 2017-02-16
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    Any upper triangular matrix that has no zeroes on the diagonal is nonsingular. Then if you want the leftmost column to not be a scalar multiple of $(1,0,0,\ldots)$ just do an elementary row operation to modify it. Elementary row operations only affect the determinant in predictable ways. – 2017-02-16
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    I am still a little confused as to how to use elementary row operations on the standard basis to get a new basis that isn't a multiple of the standard one. – 2017-02-16
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    If you add $(1,0,...,0)$ and $(0,1,0,\dots,0)$, then.... – 2017-02-16